Content-Type: multipart/mixed; boundary="-------------0003081043908" This is a multi-part message in MIME format. ---------------0003081043908 Content-Type: text/plain; name="00-103.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-103.keywords" stochastic mechanics, Stern-Gerlach, stochastic area ---------------0003081043908 Content-Type: application/postscript; name="angular.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="angular.ps" %!PS-Adobe-3.0 %%Title: (revision 1/2000) %%Creator: (Microsoft Word: LaserWriter 8 T1-8.4.1) %%CreationDate: (17:24 Mercoled\223, 8 marzo 2000) %%For: (defalco) %%Pages: 21 %%DocumentFonts: Times-Roman Times-Bold MT-Extra Times-Italic Symbol %%DocumentNeededFonts: Times-Roman Times-Bold MT-Extra Times-Italic Symbol %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 595 842 0 () () %ADO_ImageableArea: 17 22 577 819 %%EndComments userdict/dscInfo 5 dict dup begin /Title(revision 1/2000)def /Creator(Microsoft Word: LaserWriter 8 T1-8.4.1)def /CreationDate(17:24 Mercoled\223, 8 marzo 2000)def /For(defalco)def /Pages 21 def end put %%BeginFeature: *JobPatchFile 1 save /version23-manualfeedpatch where { pop false } { true }ifelse % we don't do an explicit 'get' since product and version MAY % be in systemdict or statusdict - this technique gets the lookup % without failure statusdict begin product (LaserWriter) eq % true if LaserWriter version cvr 23.0 eq % true if version 23 end and % only install this patch if both are true and % true only if patch is not installed and is for this printer % save object and boolean on stack dup { exch restore }if % either true OR saveobject false dup { /version23-manualfeedpatch true def /oldversion23-showpage /showpage load def /showpage % this showpage will wait extra time if manualfeed is true {% statusdict /manualfeed known {% manualfeed known in statusdict statusdict /manualfeed get {% if true then we loop for 5 seconds usertime 5000 add % target usertime { % loop dup usertime sub 0 lt { exit }if }loop pop % pop the usertime off the stac }if }if oldversion23-showpage }bind def }if not{ restore }if %%EndFeature /md 371 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: lw8_basic-2.0 %%Copyright: Copyright 1990-1996 Adobe Systems Incorporated and Apple Computer Incorporated. All Rights Reserved. /bd{bind def}bind def /xdf{exch def}bd /xs{exch store}bd /ld{load def}bd /Z{0 def}bd /T/true /F/false /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 14{ld}repeat /framewidth -1 def /QDframwid -1 def /numframes Z /mTS matrix def /$m matrix def /av 84 def /por T def /normland F def /psb-nosave{}def /pse-nosave{}def /us Z /psb{/us save store}bd /pse{us restore}bd /level2 /languagelevel where { pop languagelevel 2 ge }{ F }ifelse def /odictstk Z /oopstk Z /fcl { count oopstk sub dup 0 gt { {pop}repeat }{ pop }ifelse countdictstack odictstk sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /sfcl2 { /odictstk countdictstack store count/oopstk xs }bd /efcl2 { stopped{$error/newerror F put}if fcl }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /setcopies { userdict/#copies 3 -1 roll put }bd /devg/DeviceGray def /devr/DeviceRGB def /devc/DeviceCMYK def /ststpgdev{}def /dopgdev{}def /stpgdev{}def /buf Z /didstop T def /sfcl { /didstop T store /odictstk countdictstack store count/oopstk xs currentfile cvx stopped { $error/newerror F put didstop { save/didstop xs /buf vmstatus exch sub exch pop dup 0 lt{pop 0}if dup 64000 gt{pop 64000}if string store { currentfile buf readline { (}efcl)eq{exit}if }{ /UnexpectedEOF errordict/rangecheck get exec }ifelse }loop didstop restore }if }if fcl }bd /efcl { /didstop F store exec stop }bd level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /setcopies { 1 dict begin/NumCopies xdf currentdict end setpagedevice }bd /devg[/DeviceGray]def /devr[/DeviceRGB]def /devc[/DeviceCMYK]def /setpagedevice where{pop/realstpgdev/setpagedevice ld}if /SC_topddict Z /SC_spdict Z /dopgdev { md/setpagedevice undef SC_topddict realstpgdev }bd /stpgdev { SC_topddict dup 3 -1 roll { SC_spdict 2 index known { SC_spdict 2 index get dup 3 -1 roll { put dup }forall pop put dup }{ put dup }ifelse }forall pop pop }bd /ststpgdev { md/setpagedevice/stpgdev load put /SC_topddict 0 dict store /SC_spdict 3 dict begin /InputAttributes 0 dict def /Policies 0 dict def /OutputAttributes 0 dict def currentdict end store }def /sfcl/sfcl2 ld /efcl/efcl2 ld level2 not endnoload /pm Z /mT Z /sD Z /realshowpage Z /initializepage { mT $m currentmatrix mTS concatmatrix pop /pm save store mT concat }bd /endp { pm restore }bd /adjRect { dup 2 mul 6 2 roll 4 index sub exch 5 -1 roll sub exch 4 2 roll 4 index add exch 5 -1 roll add exch 4 2 roll }bd /frame1up { gS mTS setmatrix QDframwid lw /setstrokeadjust where{pop T setstrokeadjust}if clippath pathbbox 2 index sub exch 3 index sub exch currentlinewidth framewidth mul adjRect numframes dup 0 lt{pop 0}if { 4 copy rS currentlinewidth framewidth mul 4 mul adjRect }repeat pop pop pop pop gR }bd /$c devr def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile %%BeginFile: lw8_level1_colorspace-2.0 /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /$cs Z /:rg{devr :ss}bd /:sc{$cs :ss}bd /:dc { dup type/arraytype eq{0 get}if dup/DeviceCMYK eq { pop devc }{ /DeviceGray eq { devg }{ devr }ifelse }ifelse /$cs xdf }bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd %%EndFile level2 startnoload %%BeginFile: lw8_level1_basicimages-2.0 /$i F def /flipinvert statusdict begin version cvr 47.0 lt end def /iw Z /ih Z /im_save Z /setupimageproc Z /polarity Z /smoothflag Z /$z Z /bpc Z /smooth_moredata Z /datatype Z /xtraData Z /:f { /im_save save store /xtraData xs /datatype xs $i flipinvert and xor /polarity xs /smoothflag xs dup aload pop transform round 0.5 sub exch round 0.5 sub exch itransform 7 -1 roll astore concat xtraData { 0 0 1 1 1 1 idtransform abs exch pop add rC }if /$z exch string store /bpc xs xtraData{4 add}if /ih xs /iw xs /smoothflag smoothflag bpc 1 eq and smoothflag and userdict/sc_smooth known and vmstatus pop exch pop iw 3 mul sub 1000 gt and iw 4 mul 7 add 8 idiv 4 mul 65535 le and store smoothflag{ iw ih $z iw 7 add 8 idiv 4 mul string iw 4 mul 7 add 8 idiv 4 mul string T F sc_initsmooth /iw iw 4 mul store /ih ih 4 mul store }if /setupimageproc datatype 0 eq datatype 1 eq or{ smoothflag{ { [ /smooth_moredata cvx[ currentfile $z {readstring readhexstring}datatype get /pop cvx ]cvx[ $z ]cvx/ifelse cvx /sc_smooth cvx /smooth_moredata/exch cvx/store cvx ]cvx bind /smooth_moredata T store dup exec pop dup exec pop } }{ { [ currentfile $z {readstring readhexstring}datatype get /pop cvx ]cvx bind } }ifelse }{ (error, can't use level2 data acquisition procs for level1)print flush stop }ifelse store }bd /mkimgmtx { xtraData { smoothflag { [iw 0 0 ih 16 sub 0 8] }{ [iw 0 0 ih 4 sub 0 2] }ifelse }{ [iw 0 0 ih 0 0] }ifelse }bd /do1img { iw ih polarity mkimgmtx setupimageproc imagemask }bd /:j{im_save restore}bd /:g { 1 setgray 0 1 1 idtransform abs exch pop 1 1 rF 0 setgray do1img }bd /:h { :Fx 0 1 1 idtransform abs exch pop 1 1 rF :Fx do1img }bd /:i { :Fx do1img }bd %%EndFile level2 endnoload level2 not startnoload %%BeginFile: lw8_level2_basicimage-2.0 /$j 9 dict dup begin /ImageType 1 def /MultipleDataSource F def end def /im_save Z /setupimageproc Z /polarity Z /smoothflag Z /bpc Z /ih Z /iw Z /datatype Z /xtraData Z /:f { /im_save save store /xtraData xs /datatype xs datatype 0 lt datatype 8 gt or{ (error, datatype out of range)print flush stop }if /setupimageproc{ { currentfile } { currentfile 0(%ADOeod)/SubFileDecode filter/ASCIIHexDecode filter } { currentfile/RunLengthDecode filter } { currentfile/ASCII85Decode filter/RunLengthDecode filter } { currentfile/ASCII85Decode filter } { currentfile/DCTDecode filter } { currentfile/ASCII85Decode filter/DCTDecode filter } { currentfile 4 dict dup begin/Columns iw def/Rows ih def/DamagedRowsBeforeError ih def/BlackIs1 T def end/CCITTFaxDecode filter } { currentfile/ASCII85Decode filter 4 dict dup begin/Columns iw def/Rows ih def/DamagedRowsBeforeError ih def/BlackIs1 T def end/CCITTFaxDecode filter } { currentfile 5 dict dup begin/K -1 def/Columns iw def/Rows ih def/DamagedRowsBeforeError ih def/BlackIs1 T def end/CCITTFaxDecode filter } { currentfile/ASCII85Decode filter 5 dict dup begin/K -1 def/Columns iw def/Rows ih def/DamagedRowsBeforeError ih def/BlackIs1 T def end/CCITTFaxDecode filter } { currentfile 5 dict dup begin/K 1 def/Columns iw def/Rows ih def/DamagedRowsBeforeError ih def/BlackIs1 T def end/CCITTFaxDecode filter } { currentfile/ASCII85Decode filter 5 dict dup begin/K 1 def/Columns iw def/Rows ih def/DamagedRowsBeforeError ih def/BlackIs1 T def end/CCITTFaxDecode filter } }datatype get store { [1 0] }{ [0 1] }ifelse /polarity xs /smoothflag xs concat xtraData { 0 0 1 1 rC }if pop /bpc xs xtraData{4 add}if /ih xs /iw xs $c setcolorspace }bd /:j{im_save restore}bd /mkimgmtx { xtraData { [Width 0 0 Height 4 sub 0 2] }{ [Width 0 0 Height 0 0] }ifelse }bd /do1img { $j dup begin /Width iw def /Height ih def /Decode polarity def /ImageMatrix mkimgmtx def /DataSource setupimageproc def /BitsPerComponent 1 def /Interpolate smoothflag def end imagemask }bd /:g { 1 G 0 1 1 idtransform abs exch pop 1 1 rF 0 G do1img }bd /:h { :Fx 0 1 1 idtransform abs exch pop 1 1 rF :Fx do1img }bd /:i { :Fx do1img }bd %%EndFile level2 not endnoload level2 startnoload %%BeginFile: lw8_level1_colorimage-2.0 /testsystemdict { where { systemdict eq { T 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/bit3x Z /bit1x Z /concatutil { /exec load 7 -1 roll/exec load }bd /defsubclut { 1 add getinterval store }bd /spconcattransfer { /Dclut xs /Cclut xs /Bclut xs /Aclut xs /ncompute exch load store currentcolortransfer [ { Aclut ncompute } concatutil ]cvx [ { Bclut ncompute } concatutil ]cvx [ { Cclut ncompute } concatutil ]cvx [ { Dclut ncompute } concatutil ]cvx setcolortransfer }bd /setuprgbcluts { /bit3x rgbclut length 3 sub store /bit1x bit3x 3 idiv store /rclut rgbclut store /gclut rclut 1 bit3x defsubclut /bclut rclut 2 bit3x defsubclut }bd endnoload ncolors 3 ne dup startnoload /str Z /len Z /smlen Z /rstr Z /gstr Z /bstr Z /kstr Z /cmykcluttorgbclut { /str xs /len str length store /smlen len 4 idiv store /rstr str store /gstr str 1 len 1 sub getinterval store /bstr str 2 len 2 sub getinterval store /kstr str 3 len 3 sub getinterval store 0 1 smlen 1 sub { dup 3 mul/str xs 4 mul/len xs kstr len get rstr dup len get 2 index add 255 exch sub dup 0 lt{pop 0}if str exch put gstr dup len get 2 index add 255 exch sub dup 0 lt{pop 0}if str exch put bstr dup len get 2 index add 255 exch sub dup 0 lt{pop 0}if str exch put pop }for rstr 0 smlen 3 mul getinterval }bd /3compute { exch bit3x mul round cvi get 255 div }bd /:k { /rgbclut xs pop $c devc eq { rgbclut cmykcluttorgbclut/rgbclut xs }if setuprgbcluts /3compute rclut gclut bclut dup spconcattransfer iw ih bpc[iw 0 0 ih 0 0] [ setupimageproc/exec load /dup load dup ]cvx {}{}T 3 colorimage }bd endnoload ncolors 4 ne dup startnoload /cmykindex Z /bit4x Z /cmykclut Z /cclut Z /mclut Z /yclut Z /kclut Z /ftoint { 1 exch sub 255 mul round cvi }bd /stuffclut { cmykindex 3 -1 roll put }bd /4compute{exch bit4x mul round cvi get 255 div}bd /invalidcolortable? T def /computecmykclut { setuprgbcluts /bit4x rgbclut length 3 idiv 4 mul 4 sub store /cmykclut bit4x 4 add string store /cclut cmykclut store /mclut cclut 1 bit4x defsubclut /yclut cclut 2 bit4x defsubclut /kclut cclut 3 bit4x defsubclut /cmykindex 0 store 0 1 bit1x { dup/cmykindex exch bit1x exch sub 4 mul store 3 mul dup rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut }for }bd /innew Z /reverse4clut { dup length dup 4 sub/innew xdf string dup 3 -1 roll { 255 exch sub innew exch put dup innew 4 mod 3 eq { /innew innew 7 sub def }{ /innew innew 1 add def }ifelse }forall pop }bd /:k { /rgbclut xs pop $c devc ne { computecmykclut }{ /bit4x rgbclut length 4 sub store /cmykclut rgbclut reverse4clut store /cclut cmykclut store /mclut cclut 1 bit4x defsubclut /yclut cclut 2 bit4x defsubclut /kclut cclut 3 bit4x defsubclut }ifelse /4compute cclut mclut yclut kclut spconcattransfer iw ih bpc[iw 0 0 ih 0 0] [ setupimageproc/exec load /dup load dup dup ]cvx {}{}{} T 4 colorimage }bd endnoload ncolors 0 ne dup startnoload /byte Z /$x Z /padfix Z /lookupandstore { 3 mul 3 getinterval putinterval exch 3 add exch 3 copy }bd /8lookup/lookupandstore load store /4lookup { /byte 1 index store -4 bitshift lookupandstore byte 15 and lookupandstore }bd /2lookup { /byte 1 index store -6 bitshift lookupandstore byte -4 bitshift 3 and lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore }bd /colorexpand { $x 0 rgbclut 3 copy 7 -1 roll /$l load forall pop pop pop pop pop 0 padfix getinterval }bd /createexpandstr { /$x exch $z length mul string store }bd /:k { /rgbclut xs /bwclut xs /$l bpc 8 eq { 3 createexpandstr /8lookup }{ bpc 4 eq { 6 createexpandstr /4lookup }{ 12 createexpandstr /2lookup }ifelse }ifelse load store /padfix 3 iw mul store iw ih 8[iw 0 0 ih 0 0] [setupimageproc/exec load/colorexpand load/exec load]cvx F 3 colorimage }bd endnoload /colorimage where { systemdict eq { systemdict/setcmykcolor known }{ T }ifelse }{ F }ifelse dup not startnoload /:l { iw ih 8[iw 0 0 ih 0 0] setupimageproc F 3 $c devc eq { pop 4 }if colorimage }bd dup not endnoload dup startnoload /str Z /len Z /smlen Z /rstr Z /gstr Z /bstr Z /rgbtogray { /str xs /len str length store /smlen len 3 idiv store /rstr str store /gstr str 1 len 1 sub getinterval store /bstr str 2 len 2 sub getinterval store str dup 0 1 smlen 1 sub { dup 3 mul rstr 1 index get .3 mul gstr 2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put dup }for pop 0 smlen getinterval }bd /kstr Z /cmyktogray { /str xs /len str length store /smlen len 4 idiv store /rstr str store /gstr str 1 len 1 sub getinterval store /bstr str 2 len 2 sub getinterval store /kstr str 3 len 3 sub getinterval store str dup 0 1 smlen 1 sub { dup 4 mul rstr 1 index get .3 mul gstr 2 index get .59 mul add bstr 2 index get .11 mul add kstr 3 -1 roll get add round cvi 255 exch sub dup 0 lt{pop 0}if put dup }for pop 0 smlen getinterval }bd /:l { iw ih 8[iw 0 0 ih 0 0] [setupimageproc/exec load $c devc eq { /cmyktogray }{ /rgbtogray }ifelse load/exec load]cvx bind image }bd endnoload /:x { iw ih 8[iw 0 0 ih 0 0] setupimageproc image }bd %%EndFile level2 endnoload level2 not startnoload %%BeginFile: lw8_level2_colorimage-2.0 /rgbclut Z /hival Z /:k { /rgbclut xs pop bpc dup 8 eq { pop 255 }{ 4 eq { 15 }{ 3 }ifelse }ifelse /hival xs [/Indexed $c hival rgbclut]setcolorspace $j dup begin /Width iw def /Height ih def /Decode[0 hival]def /ImageMatrix[iw 0 0 ih 0 0]def /DataSource setupimageproc def /BitsPerComponent bpc def /Interpolate smoothflag def end image }bd /:l { $j dup begin /Width iw def /Height ih def /Decode[[currentcolor]length{0 1}repeat]def /ImageMatrix[iw 0 0 ih 0 0]def /DataSource setupimageproc def /BitsPerComponent 8 def /Interpolate smoothflag def end image }bd /:x { /DeviceGray setcolorspace :l }bd %%EndFile 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059629dc89b9b0ed2213e45d8a20188cae18f3391887387bf8ba0a12fe21fa0aa521bfa3 718f7abe76388e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %ADOendeexec %%EndFile %%BeginFile: lw8_uniform_graphics-2.0 /@a { np :M 0 rl :L 0 exch rl 0 rl :L fill }bd /@b { np :M 0 rl 0 exch rl :L 0 rl 0 exch rl fill }bd /@c { moveto 0 rlineto stroke }bd /@w { moveto 0 exch rlineto stroke }bd /arct where { pop }{ /arct { arcto pop pop pop pop }bd }ifelse /x1 Z /x2 Z /y1 Z /y2 Z /rad Z /@q { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct fill }bd /@s { /rad xs /y2 xs /x2 xs /y1 xs /x1 xs np x2 x1 add 2 div y1 :M x2 y1 x2 y2 rad arct x2 y2 x1 y2 rad arct x1 y2 x1 y1 rad arct x1 y1 x2 y1 rad arct :K stroke }bd /@i { np 0 360 arc fill }bd /@j { gS np :T scale 0 0 .5 0 360 arc fill gR }bd /@e { np 0 360 arc :K stroke }bd /@f { np $m currentmatrix pop :T scale 0 0 .5 0 360 arc :K $m setmatrix stroke }bd /@k { gS np :T 0 0 :M 0 0 5 2 roll arc fill gR }bd /@l { gS np :T 0 0 :M scale 0 0 .5 5 -2 roll arc fill gR }bd /@m { np arc stroke }bd /@n { np $m currentmatrix pop :T scale 0 0 .5 5 -2 roll arc $m setmatrix stroke }bd %%EndFile %%BeginFile: lw8_bubn-2.0 /$t Z /$p Z /$s Z /$o 1. def /2state? F def /ps Z level2 startnoload /pushcolor/currentrgbcolor ld /popcolor/setrgbcolor ld /setcmykcolor where { pop/currentcmykcolor where { pop/pushcolor/currentcmykcolor ld /popcolor/setcmykcolor ld }if }if level2 endnoload level2 not startnoload /pushcolor { currentcolorspace $c eq { currentcolor currentcolorspace T }{ currentcmykcolor F }ifelse }bd /popcolor { { setcolorspace setcolor }{ setcmykcolor }ifelse }bd level2 not endnoload /pushstatic { 2state? $o $t $p $s $cs ps }bd /popstatic { /ps xs /$cs xs /$s xs /$p xs /$t xs /$o xs /2state? xs }bd /pushgstate { save errordict/nocurrentpoint{pop 0 0}put currentpoint 3 -1 roll restore pushcolor currentlinewidth currentlinecap currentlinejoin currentdash exch aload length np clippath pathbbox $m currentmatrix aload pop }bd /popgstate { $m astore setmatrix 2 index sub exch 3 index sub exch rC array astore exch setdash setlinejoin setlinecap lw popcolor np :M }bd /bu { pushgstate gR pushgstate 2state? { gR pushgstate }if pushstatic pm restore mTS setmatrix }bd /bn { /pm save store popstatic popgstate gS popgstate 2state? { gS popgstate }if }bd /cpat{pop 64 div setgray 8{pop}repeat}bd %%EndFile %%BeginFile: lw8_basic_text-2.0 /S/show ld /A{ 0.0 exch ashow }bd /R{ 0.0 exch 32 exch widthshow }bd /W{ 0.0 3 1 roll widthshow }bd /J{ 0.0 32 4 2 roll 0.0 exch awidthshow }bd /V{ 0.0 4 1 roll 0.0 exch awidthshow }bd /fcflg T def /fc{ fcflg{ vmstatus exch sub 50000 lt{ (%%[ Warning: Running out of memory ]%%\r)print flush/fcflg F store }if pop }if }bd /$f[1 0 0 -1 0 0]def /:ff{$f :mf}bd /MacEncoding StandardEncoding 256 array copy def MacEncoding dup 9/space put dup 39/quotesingle put 96/grave put /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis /dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash /infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation /product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash /questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft /guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge /ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright/fi/fl /daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave /Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde /macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron MacEncoding 128 128 getinterval astore pop level2 startnoload /copyfontdict { findfont dup length dict begin { 1 index/FID ne{def}{pop pop}ifelse }forall }bd /$ckeyd md def /:skey { 1 index maxlength 2 index length sub ge { begin /$mkeys 20 dict def /$mkeys load end dup/$ckeyd xs }if 3 1 roll put }bd /fD1pass { 40 $ckeyd //md ne { pop 1 }if $ckeyd exch :skey }bd /:searchdict Z /:searchdict { exch 2 copy known { get }{ exch/$mkeys get :searchdict }ifelse }bd /lU { //md exch 2 copy known { get }{ exch /$mkeys get :searchdict }ifelse }bd level2 endnoload level2 not startnoload /copyfontdict { findfont dup length dict copy begin }bd /fD1pass/def ld /lU/load ld level2 not endnoload /fD Z /sf Z /scf Z /sf1pass { lU setfont }bd /scf1pass { exch lU exch scalefont fD }bd /scf2pass { scalefont fD }bd md/fontname known not{ /fontname/customfont def }if /Encoding Z /:mre { copyfontdict /Encoding MacEncoding def fontname currentdict end definefont :ff fD }bd /:bsr { copyfontdict /Encoding Encoding 256 array copy def Encoding dup }bd /pd{put dup}bd /:esr { pop pop fontname currentdict end definefont :ff fD }bd /ps Z /fz{/ps xs}bd /cF/currentfont ld /mbf { /makeblendedfont where { pop makeblendedfont /ABlend exch definefont }{ pop }ifelse fD }def %%EndFile %%BeginFile: lw8_derived_styles-2.0 /wi version(23.0)eq { { gS 0 0 0 0 rC stringwidth gR }bind }{ /stringwidth load }ifelse def /$o 1. def /gl{$o G}bd /ms{:M S}bd /condensedmtx[.82 0 0 1 0 0]def /:mc { condensedmtx :mf fD }bd /extendedmtx[1.18 0 0 1 0 0]def /:me { extendedmtx :mf fD }bd /basefont Z /basefonto Z /dxa Z /dxb Z /dxc Z /dxd Z /dsdx2 Z /bfproc Z /:fbase { dup/FontType get 0 eq{ dup length dict begin dup{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall /FDepVector exch/FDepVector get[exch/:fbase load forall]def }/bfproc load ifelse /customfont currentdict end definefont }bd /:mo { /bfproc{ dup dup length 2 add dict begin { 1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse }forall /PaintType 2 def /StrokeWidth .012 0 FontMatrix idtransform pop def /customfont currentdict end definefont 8 dict begin /basefonto xdf /basefont 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33<080008003F80480088008800880048003F8008000800>pd /Math :ebf %RBIEndBitmapFont /f133 f122 12 scf %%IncludeFont: MT-Extra /f154/MT-Extra findfont :ff fD /f167 f154 12 scf %%IncludeFont: Times-Italic /f178/Times-Italic :mre /f193 f178 12 scf %%IncludeFont: Symbol /f213/Symbol :bsr 240/apple pd :esr /f225 f213 12 scf /Courier findfont[10 0 0 -10 0 0]:mf setfont %%EndSetup %%Page: 1 1 %%BeginPageSetup initializepage (defalco; pag: 1 di 21)setjob %%EndPageSetup gS 0 0 560 797 rC 277 779 :M f56 sf (1)S 104 257 :M f67 sf .213 .021(Orbital Angular Momentum in Nelson's Stochastic Mechanics)J 242 284 :M f56 sf -.128(Bruno Apolloni)A 244 296 :M -.177(Diego de Falco)A 128 323 :M -.119(Dipartimento di Scienze dell'Informazione, Universit\210 di Milano)A 191 335 :M -.136(Via Comelico 39, 20135 Milano Italy)A 255 446 :M f67 sf .083(Abstract)A 53 458 :M f56 sf -.036(We provide numerical examples of integer-valued functionals of Nelson's stochastic processes.)A 53 470 :M 1.71 .171(More precisely, we consider stochastic motion, according to Nelson's form of Newton's)J 53 482 :M .323 .032(second law of dynamics, in a magnetic field having an axis )J 345 475 8 8 rC 353 483 :M -17 -23 :T psb currentpoint pse 17 23 :T 345 475 :M -17 -23 :T psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 256 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -9 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (z) show end pse gR gS 0 0 560 797 rC 353 482 :M f56 sf .767 .077( of cylindrical symmetry and a)J 53 494 :M .91 .091(gradient in the direction of this axis. We show that there are two sets of functionals of the)J 53 506 :M 1.168 .117(stochastic process having the same law as the component of quantum mechanical angular)J 53 518 :M -.166(momentum along )A 138 511 8 8 rC 146 519 :M -17 -23 :T psb currentpoint pse 17 23 :T 138 511 :M -17 -23 :T psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 256 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -9 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (z) show end pse gR gS 0 0 560 797 rC 146 518 :M f56 sf -.085(. The functionals of the first set involve the )A 352 511 8 8 rC 360 519 :M -17 -23 :T psb currentpoint pse 17 23 :T 352 511 :M -17 -23 :T psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 256 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -9 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (z) show end pse gR gS 0 0 560 797 rC 360 518 :M f56 sf .304 .03( coordinate of the process and)J 53 530 :M -.108(correctly model the behaviour of the "needle of the measuring apparatus". The functionals of the)A 53 542 :M 1.662 .166(second set involve the coordinates in a plane orthogonal to )J 364 535 8 8 rC 372 543 :M -17 -23 :T psb currentpoint pse 17 23 :T 364 535 :M -17 -23 :T psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 256 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -9 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (z) show end pse gR gS 0 0 560 797 rC 372 542 :M f56 sf 2.296 .23( and strongly suggest the)J 53 554 :M -.102(possibility of a stochastic model of the collapse of the "system" toward the state indicated by the)A 53 566 :M -.013("needle".)A endp showpage %%Page: 2 2 %%BeginPageSetup initializepage (defalco; pag: 2 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (2)S 72 97 :M f67 sf 1.166 .117(1. Introduction)J 72 115 :M f56 sf .172 .017(Of the many possible pedagogical introductions to stochastic mechanics [1], the one that best)J 72 133 :M -.055(fits with our expository strategy was given by Nelson himself in [2]. We reproduce it here with)A 72 151 :M -.107(some notational change and some minor addition.)A 72 185 :M .102 .01(Consider the stochastic process )J 226 173 63 18 rC 289 191 :M psb currentpoint pse 226 173 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2016 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate 153 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 424 moveto 224 /Times-Roman f1 (0) show 664 328 moveto 384 /Times-Italic f1 (t) show 521 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 338 moveto (\)) show 906 328 moveto 384 /Times-Roman f1 (,) show 1043 328 moveto 384 /Times-Italic f1 (t) show 1251 328 moveto 384 /Symbol f1 (\263) show 1555 328 moveto 384 /Times-Roman f1 (0) show -22 361 moveto 384 1000 1401 /Symbol f3 ({) show 1739 361 moveto (}) show end pse gR gS 17 23 560 797 rC 289 185 :M f56 sf .493 .049( defined by )J 348 174 104 16 rC 452 190 :M psb currentpoint pse 348 174 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3328 div 512 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 209 388 moveto 224 /Times-Roman f1 (0) show 502 292 moveto 384 /Times-Italic f1 (t) show 359 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 630 302 moveto (\)) show 844 292 moveto 384 /Symbol f1 (=) show 1158 292 moveto 384 /Times-Italic f1 (X) show 1456 388 moveto 224 /Times-Roman f1 (0) show 1687 292 moveto 384 /Symbol f1 (+) show 1978 292 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 2311 292 moveto 384 /Symbol f1 (\327) show 2468 292 moveto 384 /Times-Italic f1 (W) show 2982 292 moveto (t) show 2839 302 moveto 384 1000 1171 /Symbol f3 (\() show 3110 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 452 185 :M f56 sf .997 .1(, where )J 493 175 30 12 rC 523 187 :M psb currentpoint pse 493 175 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 960 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -11 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 355 261 moveto 384 /Symbol f1 (>) show 658 261 moveto 384 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 72 204 :M f56 sf -.022(is a constant, )A 137 193 66 16 rC 203 209 :M psb currentpoint pse 137 193 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2112 div 512 3 -1 roll exch div scale currentpoint translate 64 60 translate -4 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (X) show 294 388 moveto 224 /Times-Roman f1 (0) show 538 292 moveto 384 /Times-Roman f1 (~) show 846 292 moveto 384 /Times-Italic f1 (N) show 1268 292 moveto 384 /Times-Roman f1 (0) show 1458 292 moveto 384 /Times-Roman f1 (,) show 1584 292 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 1130 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1881 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 203 204 :M f56 sf -.039( \(namely )A 248 194 16 15 rC 264 209 :M psb currentpoint pse 248 194 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 480 3 -1 roll exch div scale currentpoint translate 64 28 translate -4 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (X) show 294 388 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 264 204 :M f56 sf .294 .029( is a normal random variable of mean )J 450 194 9 12 rC 459 206 :M psb currentpoint pse 450 194 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 384 3 -1 roll exch div scale currentpoint translate -1213 28 translate 1268 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 459 204 :M f56 sf ( and variance)S 72 215 12 9 rC 84 224 :M psb currentpoint pse 72 215 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 384 div 288 3 -1 roll exch div scale currentpoint translate -1535 -68 translate 1584 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show end pse gR gS 17 23 560 797 rC 84 222 :M f56 sf 2.272 .227(\) and )J 117 210 67 17 rC 184 227 :M psb currentpoint pse 117 210 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2144 div 544 3 -1 roll exch div scale currentpoint translate -2881 60 translate 3104 324 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (W) show 3618 324 moveto (t) show 3475 334 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 3746 334 moveto (\)) show 3884 324 moveto 384 /Times-Roman f1 ( ) show 3988 324 moveto 384 /Times-Roman f1 (,) show 4125 324 moveto 384 /Times-Italic f1 (t) show 4333 324 moveto 384 /Symbol f1 (\263) show 4637 324 moveto 384 /Times-Roman f1 (0) show 2923 355 moveto 384 1000 1377 /Symbol f3 ({) show 4821 355 moveto (}) show end pse gR gS 17 23 560 797 rC 184 222 :M f56 sf 2.173 .217( is a standard Brownian motion \(namely, a normal process with)J 72 229 48 15 rC 120 244 :M psb currentpoint pse 72 229 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1536 div 480 3 -1 roll exch div scale currentpoint translate -131 -5 translate 192 357 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (W) show 701 357 moveto 384 /Times-Roman f1 (0) show 563 367 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 904 367 moveto (\)) show 1118 357 moveto 384 /Symbol f1 (=) show 1427 357 moveto 384 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 120 240 :M f56 sf -.086(, and covariance function given by )A 286 228 140 17 rC 426 245 :M psb currentpoint pse 286 228 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4480 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 324 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (cov) show 688 324 moveto 384 /Times-Italic f1 (W) show 1187 324 moveto (s) show 1059 334 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1341 334 moveto (\)) show 1479 324 moveto 384 /Times-Roman f1 ( ) show 1583 324 moveto 384 /Times-Roman f1 (,) show 1724 324 moveto 384 /Times-Roman f1 ( ) show 1820 324 moveto 384 /Times-Italic f1 (W) show 2334 324 moveto (t) show 2191 334 moveto 384 1000 1171 /Symbol f3 (\() show 2462 334 moveto (\)) show 544 352 moveto 384 1000 1364 /Symbol f3 (\() show 2589 352 moveto (\)) show 2803 324 moveto 384 /Symbol f1 (=) show 3117 324 moveto 384 /Times-Roman f1 (min) show 3849 324 moveto 384 /Times-Italic f1 (s) show 3990 324 moveto 384 /Times-Roman f1 (,) show 4127 324 moveto 384 /Times-Italic f1 (t) show 3721 334 moveto 384 1000 1171 /Symbol f3 (\() show 4255 334 moveto (\)) show end pse gR gS 17 23 560 797 rC 426 240 :M f56 sf 1.007(\).)A 70 277 :M 2.687 .269(Define, for diffusion processes )J 240 266 21 15 rC 261 281 :M psb currentpoint pse 240 266 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate -3645 -2536 translate 3700 2888 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 4046 2888 moveto (t) show 3903 2898 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 4174 2898 moveto (\)) show end pse gR gS 17 23 560 797 rC 261 277 :M f56 sf 2.724 .272( with constant and assigned diffusion coefficient)J 70 282 162 23 rC 232 305 :M psb currentpoint pse 70 282 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5184 div 736 3 -1 roll exch div scale currentpoint translate 64 -2440 translate -11 2888 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 278 2717 moveto 224 /Times-Roman f1 (2) show 523 2888 moveto 384 /Symbol f1 (\272) show 897 2888 moveto 384 /Times-Roman f1 (lim) show 839 3127 moveto 224 /Times-Italic f1 (h) show 970 3127 moveto 224 /Symbol f1 (\256) show 1211 3127 moveto 224 /Times-Roman f1 (0) show 1348 3028 moveto 160 /Symbol f1 (+) show 1526 2888 moveto 384 /Times-Italic f1 (E) show 1785 2984 moveto 224 ns (t) show 2156 2888 moveto 384 ns (q) show 2502 2888 moveto (t) show 2702 2888 moveto 384 /Symbol f1 (+) show 3001 2888 moveto 384 /Times-Italic f1 (h) show 2359 2898 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 3210 2898 moveto (\)) show 3408 2888 moveto 384 /Symbol f1 (-) show 3700 2888 moveto 384 /Times-Italic f1 (q) show 4046 2888 moveto (t) show 3903 2898 moveto 384 1000 1171 /Symbol f3 (\() show 4174 2898 moveto (\)) show 2018 2916 moveto 384 1000 1364 /Symbol f3 (\() show 4301 2916 moveto (\)) show 4433 2659 moveto 224 /Times-Roman f1 (2) show 4768 2888 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -163 508 4764 2507 vb 1891 2965 moveto 384 1000 1895 /Symbol f3 (\() show 4977 2965 moveto (\)) show end pse gR gS 17 23 560 797 rC 232 296 :M f56 sf -.044(, the mean forward velocity and the mean backward velocity,)A 70 320 :M -.153(respectively by:)A 211 324 170 32 rC 381 356 :M psb currentpoint pse 211 324 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5440 div 1024 3 -1 roll exch div scale currentpoint translate 64 -1855 translate -3 2463 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 334 2463 moveto (t) show 449 2463 moveto 384 /Times-Roman f1 (,) show 581 2463 moveto 384 /Times-Italic f1 (q) show 927 2463 moveto (t) show 784 2473 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1055 2473 moveto (\)) show 191 2491 moveto 384 1000 1364 /Symbol f3 (\() show 1182 2491 moveto (\)) show 1395 2463 moveto 384 /Symbol f1 (\272) show 1769 2463 moveto 384 /Times-Roman f1 (lim) show 1711 2702 moveto 224 /Times-Italic f1 (h) show 1842 2702 moveto 224 /Symbol f1 (\256) show 2083 2702 moveto 224 /Times-Roman f1 (0) show 2220 2603 moveto 160 /Symbol f1 (+) show 2398 2463 moveto 384 /Times-Italic f1 (E) show 2657 2559 moveto 224 ns (t) show 2990 2224 moveto 384 ns (q) show 3336 2224 moveto (t) show 3536 2224 moveto 384 /Symbol f1 (+) show 3835 2224 moveto 384 /Times-Italic f1 (h) show 3193 2234 moveto 384 1000 1171 /Symbol f3 (\() show 4044 2234 moveto (\)) show 4242 2224 moveto 384 /Symbol f1 (-) show 4534 2224 moveto 384 /Times-Italic f1 (q) show 4880 2224 moveto (t) show 4737 2234 moveto 384 1000 1171 /Symbol f3 (\() show 5008 2234 moveto (\)) show 3961 2756 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2967 2364 moveto 2179 0 rlineto stroke 2768 2255 moveto 384 /Symbol f1 (\346) show 2768 2716 moveto (\350) show 2768 2497 moveto (\347) show 5158 2255 moveto (\366) show 5158 2716 moveto (\370) show 5158 2497 moveto (\367) show end pse gR gS 209 375 175 32 rC 384 407 :M psb currentpoint pse 209 375 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5600 div 1024 3 -1 roll exch div scale currentpoint translate 64 -1855 translate -3 2463 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 197 2292 moveto 224 /Times-Roman f1 (*) show 481 2463 moveto 384 /Times-Italic f1 (t) show 596 2463 moveto 384 /Times-Roman f1 (,) show 728 2463 moveto 384 /Times-Italic f1 (q) show 1074 2463 moveto (t) show 931 2473 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1202 2473 moveto (\)) show 338 2491 moveto 384 1000 1364 /Symbol f3 (\() show 1329 2491 moveto (\)) show 1542 2463 moveto 384 /Symbol f1 (\272) show 1916 2463 moveto 384 /Times-Roman f1 (lim) show 1858 2702 moveto 224 /Times-Italic f1 (h) show 1989 2702 moveto 224 /Symbol f1 (\256) show 2230 2702 moveto 224 /Times-Roman f1 (0) show 2367 2603 moveto 160 /Symbol f1 (+) show 2545 2463 moveto 384 /Times-Italic f1 (E) show 2804 2559 moveto 224 ns (t) show 3137 2224 moveto 384 ns (q) show 3483 2224 moveto (t) show 3340 2234 moveto 384 1000 1171 /Symbol f3 (\() show 3611 2234 moveto (\)) show 3809 2224 moveto 384 /Symbol f1 (-) show 4101 2224 moveto 384 /Times-Italic f1 (q) show 4447 2224 moveto (t) show 4646 2224 moveto 384 /Symbol f1 (-) show 4944 2224 moveto 384 /Times-Italic f1 (h) show 4304 2234 moveto 384 1000 1171 /Symbol f3 (\() show 5153 2234 moveto (\)) show 4107 2756 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3114 2364 moveto 2177 0 rlineto stroke 2915 2255 moveto 384 /Symbol f1 (\346) show 2915 2716 moveto (\350) show 2915 2497 moveto (\347) show 5303 2255 moveto (\366) show 5303 2716 moveto (\370) show 5303 2497 moveto (\367) show end pse gR gS 17 23 560 797 rC 70 420 :M f56 sf -.06(\(Here )A 100 410 13 15 rC 113 425 :M psb currentpoint pse 100 410 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate -2338 -2834 translate 2398 3154 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (E) show 2657 3250 moveto 224 ns (t) show end pse gR gS 17 23 560 797 rC 113 420 :M f56 sf -.15( indicates the conditional expectation given the value of the process at time )A 465 412 7 10 rC 472 422 :M psb currentpoint pse 465 412 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -274 -2898 translate 334 3154 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 472 420 :M f56 sf 1.007(\).)A 70 456 :M .343 .034(The above limits are easily shown to exist for the process )J 354 445 26 16 rC 380 461 :M psb currentpoint pse 354 445 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate -98 -1876 translate 153 2228 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 2324 moveto 224 /Times-Roman f1 (0) show 664 2228 moveto 384 /Times-Italic f1 (t) show 521 2238 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 2238 moveto (\)) show end pse gR gS 17 23 560 797 rC 380 456 :M f56 sf .247 .025(: they are given, respectively,)J 70 476 :M -.167(by:)A 260 480 73 18 rC 333 498 :M psb currentpoint pse 260 480 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2336 div 576 3 -1 roll exch div scale currentpoint translate 64 -2535 translate -3 2919 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 206 3015 moveto 224 /Times-Roman f1 (0) show 499 2919 moveto 384 /Times-Italic f1 (t) show 614 2919 moveto 384 /Times-Roman f1 (,) show 746 2919 moveto 384 /Times-Italic f1 (q) show 964 3015 moveto 224 /Times-Roman f1 (0) show 1257 2919 moveto 384 /Times-Italic f1 (t) show 1114 2929 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1385 2929 moveto (\)) show 356 2949 moveto 384 1000 1388 /Symbol f3 (\() show 1512 2949 moveto (\)) show 1726 2919 moveto 384 /Symbol f1 (=) show 2035 2919 moveto 384 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 70 513 :M f56 sf -.163(and)A 232 536 :M ( )S 238 517 113 30 rC 351 547 :M psb currentpoint pse 238 517 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3616 div 960 3 -1 roll exch div scale currentpoint translate 64 -2545 translate -3 3153 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 206 3249 moveto 224 /Times-Roman f1 (0) show 197 2982 moveto 224 /Times-Roman f1 (*) show 499 3153 moveto 384 /Times-Italic f1 (t) show 614 3153 moveto 384 /Times-Roman f1 (,) show 746 3153 moveto 384 /Times-Italic f1 (q) show 964 3249 moveto 224 /Times-Roman f1 (0) show 1257 3153 moveto 384 /Times-Italic f1 (t) show 1114 3163 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1385 3163 moveto (\)) show 356 3183 moveto 384 1000 1388 /Symbol f3 (\() show 1512 3183 moveto (\)) show 1726 3153 moveto 384 /Symbol f1 (=) show 2076 2914 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 2365 2743 moveto 224 /Times-Roman f1 (2) show 2573 2914 moveto 384 /Symbol f1 (\327) show 2724 2914 moveto 384 /Times-Italic f1 (q) show 2942 3010 moveto 224 /Times-Roman f1 (0) show 3235 2914 moveto 384 /Times-Italic f1 (t) show 3092 2924 moveto 384 1000 1171 /Symbol f3 (\() show 3363 2924 moveto (\)) show 2061 3446 moveto 384 /Symbol f2 (a) show 2430 3446 moveto 384 /Symbol f1 (+) show 2721 3446 moveto 384 /Symbol f2 (s) show 3010 3275 moveto 224 /Times-Roman f1 (2) show 3218 3446 moveto 384 /Symbol f1 (\327) show 3374 3446 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2044 3054 moveto 1469 0 rlineto stroke end pse gR gS 17 23 560 797 rC 351 536 :M f56 sf .455 .045( .)J 70 578 :M -.053(Having set, more generally, for a function )A 272 566 45 17 rC 317 583 :M psb currentpoint pse 272 566 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1440 div 544 3 -1 roll exch div scale currentpoint translate -210 -2138 translate 270 2522 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (F) show 633 2522 moveto (t) show 748 2522 moveto 384 /Times-Roman f1 (,) show 880 2522 moveto 384 /Times-Italic f1 (q) show 1226 2522 moveto (t) show 1083 2532 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1354 2532 moveto (\)) show 490 2550 moveto 384 1000 1364 /Symbol f3 (\() show 1481 2550 moveto (\)) show end pse gR gS 17 23 560 797 rC 317 578 :M f56 sf (,)S 174 584 244 36 rC 418 620 :M psb currentpoint pse 174 584 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7808 div 1152 3 -1 roll exch div scale currentpoint translate 64 -1850 translate -7 2522 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (DF) show 633 2522 moveto (t) show 748 2522 moveto 384 /Times-Roman f1 (,) show 880 2522 moveto 384 /Times-Italic f1 (q) show 1226 2522 moveto (t) show 1083 2532 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1354 2532 moveto (\)) show 490 2550 moveto 384 1000 1364 /Symbol f3 (\() show 1481 2550 moveto (\)) show 1695 2522 moveto 384 /Symbol f1 (=) show 2069 2522 moveto 384 /Times-Roman f1 (lim) show 2011 2761 moveto 224 /Times-Italic f1 (h) show 2142 2761 moveto 224 /Symbol f1 (\256) show 2383 2761 moveto 224 /Times-Roman f1 (0) show 2520 2662 moveto 160 /Symbol f1 (+) show 2698 2522 moveto 384 /Times-Italic f1 (E) show 2957 2618 moveto 224 ns (t) show 3295 2256 moveto 384 ns (F) show 3658 2256 moveto (t) show 3858 2256 moveto 384 /Symbol f1 (+) show 4157 2256 moveto 384 /Times-Italic f1 (h) show 4353 2256 moveto 384 /Times-Roman f1 (,) show 4485 2256 moveto 384 /Times-Italic f1 (q) show 4831 2256 moveto (t) show 5031 2256 moveto 384 /Symbol f1 (+) show 5330 2256 moveto 384 /Times-Italic f1 (h) show 4688 2266 moveto 384 1000 1171 /Symbol f3 (\() show 5539 2266 moveto (\)) show 3515 2284 moveto 384 1000 1364 /Symbol f3 (\() show 5666 2284 moveto (\)) show 5864 2256 moveto 384 /Symbol f1 (-) show 6161 2256 moveto 384 /Times-Italic f1 (F) show 6524 2256 moveto (t) show 6639 2256 moveto 384 /Times-Roman f1 (,) show 6771 2256 moveto 384 /Times-Italic f1 (q) show 7117 2256 moveto (t) show 6974 2266 moveto 384 1000 1171 /Symbol f3 (\() show 7245 2266 moveto (\)) show 6381 2284 moveto 384 1000 1364 /Symbol f3 (\() show 7372 2284 moveto (\)) show 5293 2815 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3267 2423 moveto 4243 0 rlineto stroke 3068 2255 moveto 384 /Symbol f1 (\346) show 3068 2834 moveto (\350) show 3068 2615 moveto (\347) show 7522 2255 moveto (\366) show 7522 2834 moveto (\370) show 7522 2615 moveto (\367) show end pse gR gS 172 639 248 36 rC 420 675 :M psb currentpoint pse 172 639 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7936 div 1152 3 -1 roll exch div scale currentpoint translate 64 -1850 translate -7 2522 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (D) show 275 2351 moveto 224 /Times-Roman f1 (*) show 400 2522 moveto 384 /Times-Italic f1 (F) show 763 2522 moveto (t) show 878 2522 moveto 384 /Times-Roman f1 (,) show 1010 2522 moveto 384 /Times-Italic f1 (q) show 1356 2522 moveto (t) show 1213 2532 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1484 2532 moveto (\)) show 620 2550 moveto 384 1000 1364 /Symbol f3 (\() show 1611 2550 moveto (\)) show 1824 2522 moveto 384 /Symbol f1 (\272) show 2198 2522 moveto 384 /Times-Roman f1 (lim) show 2140 2761 moveto 224 /Times-Italic f1 (h) show 2271 2761 moveto 224 /Symbol f1 (\256) show 2512 2761 moveto 224 /Times-Roman f1 (0) show 2649 2662 moveto 160 /Symbol f1 (+) show 2827 2522 moveto 384 /Times-Italic f1 (E) show 3086 2618 moveto 224 ns (t) show 3424 2256 moveto 384 ns (F) show 3787 2256 moveto (t) show 3902 2256 moveto 384 /Times-Roman f1 (,) show 4034 2256 moveto 384 /Times-Italic f1 (q) show 4380 2256 moveto (t) show 4237 2266 moveto 384 1000 1171 /Symbol f3 (\() show 4508 2266 moveto (\)) show 3644 2284 moveto 384 1000 1364 /Symbol f3 (\() show 4635 2284 moveto (\)) show 4833 2256 moveto 384 /Symbol f1 (-) show 5130 2256 moveto 384 /Times-Italic f1 (F) show 5493 2256 moveto (t) show 5692 2256 moveto 384 /Symbol f1 (-) show 5990 2256 moveto 384 /Times-Italic f1 (h) show 6186 2256 moveto 384 /Times-Roman f1 (,) show 6318 2256 moveto 384 /Times-Italic f1 (q) show 6664 2256 moveto (t) show 6863 2256 moveto 384 /Symbol f1 (-) show 7161 2256 moveto 384 /Times-Italic f1 (h) show 6521 2266 moveto 384 1000 1171 /Symbol f3 (\() show 7370 2266 moveto (\)) show 5350 2284 moveto 384 1000 1364 /Symbol f3 (\() show 7497 2284 moveto (\)) show 5420 2815 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3396 2423 moveto 4239 0 rlineto stroke 3197 2255 moveto 384 /Symbol f1 (\346) show 3197 2834 moveto (\350) show 3197 2615 moveto (\347) show 7647 2255 moveto (\366) show 7647 2834 moveto (\370) show 7647 2615 moveto (\367) show end pse gR gS 17 23 560 797 rC 70 690 :M f56 sf -.114(one can inquire about the mean acceleration of the process, defined by)A 206 712 177 31 rC 383 743 :M psb currentpoint pse 206 712 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5664 div 992 3 -1 roll exch div scale currentpoint translate 64 -4350 translate -14 4990 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (a) show 310 4990 moveto (t) show 425 4990 moveto 384 /Times-Roman f1 (,) show 557 4990 moveto 384 /Times-Italic f1 (q) show 903 4990 moveto (t) show 760 5000 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1031 5000 moveto (\)) show 167 5018 moveto 384 1000 1364 /Symbol f3 (\() show 1158 5018 moveto (\)) show 1372 4990 moveto 384 /Symbol f1 (=) show 1715 4724 moveto 384 /Times-Italic f1 (D) show 1997 4553 moveto 224 /Times-Roman f1 (*) show 2123 4724 moveto 384 /Times-Italic f1 (b) show 2460 4724 moveto (t) show 2575 4724 moveto 384 /Times-Roman f1 (,) show 2707 4724 moveto 384 /Times-Italic f1 (q) show 3053 4724 moveto (t) show 2910 4734 moveto 384 1000 1171 /Symbol f3 (\() show 3181 4734 moveto (\)) show 2317 4752 moveto 384 1000 1364 /Symbol f3 (\() show 3308 4752 moveto (\)) show 3507 4724 moveto 384 /Symbol f1 (+) show 3802 4724 moveto 384 /Times-Italic f1 (Db) show 4279 4553 moveto 224 /Times-Roman f1 (*) show 4563 4724 moveto 384 /Times-Italic f1 (t) show 4678 4724 moveto 384 /Times-Roman f1 (,) show 4810 4724 moveto 384 /Times-Italic f1 (q) show 5156 4724 moveto (t) show 5013 4734 moveto 384 1000 1171 /Symbol f3 (\() show 5284 4734 moveto (\)) show 4420 4752 moveto 384 1000 1364 /Symbol f3 (\() show 5411 4752 moveto (\)) show 3523 5283 moveto 384 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1690 4891 moveto 3859 0 rlineto stroke end pse gR gS 17 23 560 797 rC 383 732 :M f56 sf (.)S endp showpage %%Page: 3 3 %%BeginPageSetup initializepage (defalco; pag: 3 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (3)S 70 97 :M .28 .028(The limit defining the above quantity is easily shown to exist for the process )J 446 86 26 16 rC 472 102 :M psb currentpoint pse 446 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate -98 -1876 translate 153 2228 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 2324 moveto 224 /Times-Roman f1 (0) show 664 2228 moveto 384 /Times-Italic f1 (t) show 521 2238 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 2238 moveto (\)) show end pse gR gS 17 23 560 797 rC 472 97 :M f56 sf 1.029 .103( and to be)J 70 117 :M -.175(given explicitly by:)A 223 121 146 36 rC 369 157 :M psb currentpoint pse 223 121 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4672 div 1152 3 -1 roll exch div scale currentpoint translate 64 -4354 translate -14 4962 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (a) show 182 5058 moveto 224 /Times-Roman f1 (0) show 475 4962 moveto 384 /Times-Italic f1 (t) show 590 4962 moveto 384 /Times-Roman f1 (,) show 722 4962 moveto 384 /Times-Italic f1 (q) show 940 5058 moveto 224 /Times-Roman f1 (0) show 1233 4962 moveto 384 /Times-Italic f1 (t) show 1090 4972 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1361 4972 moveto (\)) show 332 4992 moveto 384 1000 1388 /Symbol f3 (\() show 1488 4992 moveto (\)) show 1702 4962 moveto 384 /Symbol f1 (=) show 2016 4962 moveto (-) show 2726 4723 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 3019 4552 moveto 224 /Times-Roman f1 (4) show 3226 4723 moveto 384 /Symbol f1 (\327) show 3377 4723 moveto 384 /Times-Italic f1 (q) show 3595 4819 moveto 224 /Times-Roman f1 (0) show 3888 4723 moveto 384 /Times-Italic f1 (t) show 3745 4733 moveto 384 1000 1171 /Symbol f3 (\() show 4016 4733 moveto (\)) show 2306 5314 moveto 384 /Times-Roman f1 (2) show 2558 5314 moveto 384 /Symbol f1 (\327) show 2830 5314 moveto 384 /Symbol f2 (a) show 3199 5314 moveto 384 /Symbol f1 (+) show 3490 5314 moveto 384 /Symbol f2 (s) show 3779 5143 moveto 224 /Times-Roman f1 (2) show 3987 5314 moveto 384 /Symbol f1 (\327) show 4143 5314 moveto 384 /Times-Italic f1 (t) show 2698 5358 moveto 384 1000 1544 /Symbol f3 (\() show 4271 5358 moveto (\)) show 4403 5055 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2285 4863 moveto 2288 0 rlineto stroke end pse gR gS 17 23 560 797 rC 70 188 :M f56 sf -.058(The process )A 130 177 26 16 rC 156 193 :M psb currentpoint pse 130 177 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate -98 -1876 translate 153 2228 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 2324 moveto 224 /Times-Roman f1 (0) show 664 2228 moveto 384 /Times-Italic f1 (t) show 521 2238 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 2238 moveto (\)) show end pse gR gS 17 23 560 797 rC 156 188 :M f56 sf -.049( \(namely Brownian motion with initial condition )A 390 177 39 15 rC 429 192 :M psb currentpoint pse 390 177 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1248 div 480 3 -1 roll exch div scale currentpoint translate -786 -4340 translate 846 4692 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (N) show 1268 4692 moveto 384 /Times-Roman f1 (0) show 1458 4692 moveto 384 /Times-Roman f1 (,) show 1584 4692 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 1130 4702 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1881 4702 moveto (\)) show end pse gR gS 17 23 560 797 rC 429 188 :M f56 sf .22 .022( \) appears therefore)J 70 207 :M .659 .066(as a particular solution of the problem of finding the diffusion processes )J 432 196 21 15 rC 453 211 :M psb currentpoint pse 432 196 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate -3645 -2536 translate 3700 2888 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 4046 2888 moveto (t) show 3903 2898 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 4174 2898 moveto (\)) show end pse gR gS 17 23 560 797 rC 453 207 :M f56 sf .79 .079( satisfying the)J 70 226 :M -.133(conditions:)A 70 298 :M ( )S 208 248 195 94 rC 403 342 :M psb currentpoint pse 208 248 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6240 div 3008 3 -1 roll exch div scale currentpoint translate 64 -4364 translate 257 5111 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (lim) show 199 5350 moveto 224 /Times-Italic f1 (h) show 330 5350 moveto 224 /Symbol f1 (\256) show 571 5350 moveto 224 /Times-Roman f1 (0) show 708 5251 moveto 160 /Symbol f1 (+) show 886 5111 moveto 384 /Times-Italic f1 (E) show 1145 5207 moveto 224 ns (t) show 1605 4845 moveto 384 ns (q) show 1951 4845 moveto (t) show 2151 4845 moveto 384 /Symbol f1 (+) show 2450 4845 moveto 384 /Times-Italic f1 (h) show 1808 4855 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 2659 4855 moveto (\)) show 2857 4845 moveto 384 /Symbol f1 (-) show 3149 4845 moveto 384 /Times-Italic f1 (q) show 3495 4845 moveto (t) show 3352 4855 moveto 384 1000 1171 /Symbol f3 (\() show 3623 4855 moveto (\)) show 1467 4873 moveto 384 1000 1364 /Symbol f3 (\() show 3750 4873 moveto (\)) show 3882 4616 moveto 224 /Times-Roman f1 (2) show 2658 5404 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1455 5012 moveto 2597 0 rlineto stroke 1256 4787 moveto 384 /Symbol f1 (\346) show 1256 5480 moveto (\350) show 1256 5238 moveto (\347) show 4064 4787 moveto (\366) show 4064 5480 moveto (\370) show 4064 5238 moveto (\367) show 4344 5111 moveto (=) show 4651 5111 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 4940 4940 moveto 224 /Times-Roman f1 (2) show 5078 5111 moveto 384 /Times-Roman f1 ( ) show 220 6551 moveto 384 /Times-Italic f1 (D) show 502 6380 moveto 224 /Times-Roman f1 (*) show 624 6551 moveto 384 /Times-Italic f1 (Dq) show 1247 6551 moveto (t) show 1104 6561 moveto 384 1000 1171 /Symbol f3 (\() show 1375 6561 moveto (\)) show 1574 6551 moveto 384 /Symbol f1 (+) show 1869 6551 moveto 384 /Times-Italic f1 (DD) show 2428 6380 moveto 224 /Times-Roman f1 (*) show 2548 6551 moveto 384 /Times-Italic f1 (q) show 2894 6551 moveto (t) show 2751 6561 moveto 384 1000 1171 /Symbol f3 (\() show 3022 6561 moveto (\)) show 1581 7083 moveto 384 /Times-Roman f1 (2) show 195 6691 moveto 2965 0 rlineto stroke 3268 6790 moveto 384 /Symbol f1 (=) show 3582 6790 moveto (-) show 4374 6551 moveto 384 /Symbol f2 (s) show 4667 6380 moveto 224 /Times-Roman f1 (4) show 4874 6551 moveto 384 /Symbol f1 (\327) show 5025 6551 moveto 384 /Times-Italic f1 (q) show 5371 6551 moveto (t) show 5228 6561 moveto 384 1000 1171 /Symbol f3 (\() show 5499 6561 moveto (\)) show 3872 7142 moveto 384 /Times-Roman f1 (2) show 4124 7142 moveto 384 /Symbol f1 (\327) show 4396 7142 moveto 384 /Symbol f2 (a) show 4765 7142 moveto 384 /Symbol f1 (+) show 5056 7142 moveto 384 /Symbol f2 (s) show 5345 6971 moveto 224 /Times-Roman f1 (2) show 5553 7142 moveto 384 /Symbol f1 (\327) show 5709 7142 moveto 384 /Times-Italic f1 (t) show 4264 7186 moveto 384 1000 1544 /Symbol f3 (\() show 5837 7186 moveto (\)) show 5969 6883 moveto 224 /Times-Roman f1 (2) show 3851 6691 moveto 2288 0 rlineto stroke -5 4755 moveto 384 /Symbol f1 (\354) show -5 6029 moveto (\355) show -5 5126 moveto (\357) show -5 5499 moveto (\357) show -5 5737 moveto (\357) show -5 7304 moveto (\356) show -5 6403 moveto (\357) show -5 6776 moveto (\357) show -5 7149 moveto (\357) show end pse gR gS 17 23 560 797 rC 403 298 :M f56 sf ( \(1\))S 70 375 :M .082 .008(As an exercise in the theory of diffusion processes one can pose the problem of finding )J f78 sf 12 u78 :p 12.052 :m .019(all)A f56 sf .066 .007( the)J 70 393 :M -.013(solutions of the problem posed by system \(1\). Such a problem is easily solved according to the)A 70 411 :M -.094(following three steps:)A 70 446 :M 1.148 .115(Step 1: Introduce the current velocity field )J 290 433 137 19 rC 427 452 :M psb currentpoint pse 290 433 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4384 div 608 3 -1 roll exch div scale currentpoint translate 64 -7500 translate -3 7916 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 331 7916 moveto (t) show 446 7916 moveto 384 /Times-Roman f1 (,) show 581 7916 moveto 384 /Times-Italic f1 (x) show 188 7926 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 785 7926 moveto (\)) show 999 7916 moveto 384 /Symbol f1 (=) show 1441 7916 moveto 384 /Times-Italic f1 (b) show 1778 7916 moveto (t) show 1893 7916 moveto 384 /Times-Roman f1 (,) show 2028 7916 moveto 384 /Times-Italic f1 (x) show 1635 7926 moveto 384 1000 1171 /Symbol f3 (\() show 2232 7926 moveto (\)) show 2431 7916 moveto 384 /Symbol f1 (+) show 2730 7916 moveto 384 /Times-Italic f1 (b) show 2930 7745 moveto 224 /Times-Roman f1 (*) show 3214 7916 moveto 384 /Times-Italic f1 (t) show 3329 7916 moveto 384 /Times-Roman f1 (,) show 3464 7916 moveto 384 /Times-Italic f1 (x) show 3071 7926 moveto 384 1000 1171 /Symbol f3 (\() show 3668 7926 moveto (\)) show 1297 7960 moveto 384 1000 1544 /Symbol f3 (\() show 3795 7960 moveto (\)) show 4100 7916 moveto 384 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -164 510 4095 7562 vb end pse gR gS 17 23 560 797 rC 427 446 :M f56 sf 1.994 .199( and ask for those)J 70 467 :M -.105(solutions of \(1\) which have the same velocity field)A 237 471 119 35 rC 356 506 :M psb currentpoint pse 237 471 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3808 div 1120 3 -1 roll exch div scale currentpoint translate 64 46 translate -3 562 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 225 658 moveto 224 /Times-Roman f1 (0) show 518 562 moveto 384 /Times-Italic f1 (t) show 661 562 moveto 384 /Times-Roman f1 (,) show 796 562 moveto 384 /Times-Italic f1 (x) show 375 572 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1023 572 moveto (\)) show 1237 562 moveto 384 /Symbol f1 (=) show 2197 323 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 2486 152 moveto 224 /Times-Roman f1 (2) show 2694 323 moveto 384 /Symbol f1 (\327) show 2848 323 moveto 384 /Times-Italic f1 (x) show 1576 857 moveto 384 /Times-Roman f1 (2) show 1828 857 moveto 384 /Symbol f1 (\327) show 2100 857 moveto 384 /Symbol f2 (a) show 2469 857 moveto 384 /Symbol f1 (+) show 2760 857 moveto 384 /Symbol f2 (s) show 3049 686 moveto 224 /Times-Roman f1 (2) show 3257 857 moveto 384 /Symbol f1 (\327) show 3413 857 moveto 384 /Times-Italic f1 (t) show 1968 901 moveto 384 1000 1544 /Symbol f3 (\() show 3569 901 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1555 463 moveto 2152 0 rlineto stroke end pse gR gS 17 23 560 797 rC 70 519 :M f56 sf -.045(as the process )A 139 508 26 16 rC 165 524 :M psb currentpoint pse 139 508 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate -98 -1876 translate 153 2228 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 2324 moveto 224 /Times-Roman f1 (0) show 664 2228 moveto 384 /Times-Italic f1 (t) show 521 2238 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 2238 moveto (\)) show end pse gR gS 17 23 560 797 rC 165 519 :M f56 sf (.)S 70 556 :M .348 .035(It is easy to show that this intermediate problem has a countable infinity )J 426 545 82 15 rC 508 560 :M psb currentpoint pse 426 545 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2624 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 234 388 moveto 224 ns (n) show 543 292 moveto 384 ns (t) show 400 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 699 302 moveto (\)) show 813 292 moveto 384 /Times-Roman f1 (,) show 954 292 moveto 384 /Times-Roman f1 ( ) show 1050 292 moveto 384 /Times-Italic f1 (n) show 1361 292 moveto 384 /Times-Roman f1 (=) show 1669 292 moveto 384 /Times-Roman f1 (0) show 1859 292 moveto 384 /Times-Roman f1 (,) show 1958 292 moveto 384 /Times-Roman f1 (1) show 2117 292 moveto 384 /Times-Roman f1 (,) show 2232 292 moveto (.) show 2340 292 moveto (.) show 2448 292 moveto (.) show end pse gR gS 17 23 560 797 rC 508 556 :M f56 sf 1.672 .167( of)J 70 574 :M -.062(solutions, the probability density of )A 241 563 27 15 rC 268 578 :M psb currentpoint pse 241 563 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 212 388 moveto 224 ns (n) show 508 292 moveto 384 ns (t) show 365 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 636 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 268 574 :M f56 sf -.094( being given by)A 169 597 254 94 rC 423 691 :M psb currentpoint pse 169 597 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8128 div 3008 3 -1 roll exch div scale currentpoint translate 64 41 translate -19 681 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (r) show 216 777 moveto 224 /Times-Italic f1 (n) show 512 681 moveto 384 ns (t) show 627 681 moveto 384 /Times-Roman f1 (,) show 762 681 moveto 384 /Times-Italic f1 (x) show 369 691 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 966 691 moveto (\)) show 1180 681 moveto 384 /Symbol f1 (=) show 2844 442 moveto 384 /Times-Roman f1 (1) show 1781 1056 moveto (2) show 2033 1056 moveto 384 /Symbol f1 (\327) show 2190 1056 moveto 384 /Symbol f2 (p) show 2500 1056 moveto 384 /Symbol f1 (\327) show 2772 1056 moveto 384 /Symbol f2 (a) show 3141 1056 moveto 384 /Symbol f1 (+) show 3432 1056 moveto 384 /Symbol f2 (s) show 3721 885 moveto 224 /Times-Roman f1 (2) show 3929 1056 moveto 384 /Symbol f1 (\327) show 4085 1056 moveto 384 /Times-Italic f1 (t) show 2640 1100 moveto 384 1000 1544 /Symbol f3 (\() show 4213 1100 moveto (\)) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 2821 622 384 1530 1244 16 sqr /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1498 582 moveto 2885 0 rlineto stroke 4453 681 moveto 384 /Symbol f1 (\327) show 4605 681 moveto 384 /Times-Roman f1 (exp) show 5359 681 moveto 384 /Symbol f1 (-) show 6514 442 moveto 384 /Times-Italic f1 (x) show 6723 271 moveto 224 /Times-Roman f1 (2) show 5649 976 moveto 384 ns (2) show 5901 976 moveto 384 /Symbol f1 (\327) show 6173 976 moveto 384 /Symbol f2 (a) show 6542 976 moveto 384 /Symbol f1 (+) show 6833 976 moveto 384 /Symbol f2 (s) show 7122 805 moveto 224 /Times-Roman f1 (2) show 7330 976 moveto 384 /Symbol f1 (\327) show 7486 976 moveto 384 /Times-Italic f1 (t) show 6041 1020 moveto 384 1000 1544 /Symbol f3 (\() show 7614 1020 moveto (\)) show 5628 582 moveto 2124 0 rlineto stroke 5164 355 moveto 384 /Symbol f1 (\346) show 5164 1052 moveto (\350) show 5164 806 moveto (\347) show 7764 355 moveto (\366) show 7764 1052 moveto (\370) show 7764 806 moveto (\367) show 7942 681 moveto 384 /Times-Roman f1 (.) show 0 2326 moveto 384 /Times-Roman f1 ( ) show 1873 2087 moveto (1) show 1525 2619 moveto 384 /Times-Roman f1 (2) show 1737 2448 moveto 224 /Times-Italic f1 (n) show 1948 2619 moveto 384 /Symbol f1 (\327) show 2105 2619 moveto 384 /Times-Italic f1 (n) show 2316 2619 moveto 384 /Times-Roman f1 (!) show 1504 2227 moveto 930 0 rlineto stroke 2504 2326 moveto 384 /Symbol f1 (\327) show 2657 2326 moveto 384 /Times-Italic f1 (H) show 2956 2422 moveto 224 ns (n) show 4428 2087 moveto 384 ns (x) show 3596 2701 moveto 384 /Times-Roman f1 (2) show 3848 2701 moveto 384 /Symbol f1 (\327) show 4120 2701 moveto 384 /Symbol f2 (a) show 4489 2701 moveto 384 /Symbol f1 (+) show 4780 2701 moveto 384 /Symbol f2 (s) show 5069 2530 moveto 224 /Times-Roman f1 (2) show 5277 2701 moveto 384 /Symbol f1 (\327) show 5433 2701 moveto 384 /Times-Italic f1 (t) show 3988 2745 moveto 384 1000 1544 /Symbol f3 (\() show 5561 2745 moveto (\)) show 2354 622 384 3345 2889 16 sqr 3313 2227 moveto 2418 0 rlineto stroke 3114 1888 moveto 384 /Symbol f1 (\346) show 3114 2809 moveto (\350) show 3114 2339 moveto (\347) show 3114 2590 moveto (\347) show 5743 1888 moveto (\366) show 5743 2809 moveto (\370) show 5743 2339 moveto (\367) show 5743 2590 moveto (\367) show 5941 1628 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 72 723 :M f56 sf -.062(where )A 104 713 16 14 rC 120 727 :M psb currentpoint pse 104 713 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 448 3 -1 roll exch div scale currentpoint translate -2600 -2006 translate 2657 2326 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 2956 2422 moveto 224 ns (n) show end pse gR gS 17 23 560 797 rC 120 723 :M f56 sf -.094( is the )A 151 713 31 12 rC 182 725 :M psb currentpoint pse 151 713 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 992 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -3 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (n) show 277 261 moveto 384 /Symbol f1 (-) show 574 261 moveto 384 /Times-Italic f1 (th) show end pse gR gS 17 23 560 797 rC 182 723 :M f56 sf -.11(Hermite polynomial.)A endp showpage %%Page: 4 4 %%BeginPageSetup initializepage (defalco; pag: 4 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (4)S 72 99 :M .368 .037(Step 2: Introduce the osmotic "velocity" field )J 297 86 137 19 rC 434 105 :M psb currentpoint pse 297 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4384 div 608 3 -1 roll exch div scale currentpoint translate 64 62 translate -3 354 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (u) show 335 354 moveto (t) show 450 354 moveto 384 /Times-Roman f1 (,) show 585 354 moveto 384 /Times-Italic f1 (x) show 192 364 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 789 364 moveto (\)) show 1003 354 moveto 384 /Symbol f1 (=) show 1445 354 moveto 384 /Times-Italic f1 (b) show 1782 354 moveto (t) show 1897 354 moveto 384 /Times-Roman f1 (,) show 2032 354 moveto 384 /Times-Italic f1 (x) show 1639 364 moveto 384 1000 1171 /Symbol f3 (\() show 2236 364 moveto (\)) show 2434 354 moveto 384 /Symbol f1 (-) show 2732 354 moveto 384 /Times-Italic f1 (b) show 2932 183 moveto 224 /Times-Roman f1 (*) show 3216 354 moveto 384 /Times-Italic f1 (t) show 3331 354 moveto 384 /Times-Roman f1 (,) show 3466 354 moveto 384 /Times-Italic f1 (x) show 3073 364 moveto 384 1000 1171 /Symbol f3 (\() show 3670 364 moveto (\)) show 1301 398 moveto 384 1000 1544 /Symbol f3 (\() show 3797 398 moveto (\)) show 4102 354 moveto 384 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -164 510 4097 0 vb end pse gR gS 17 23 560 797 rC 434 99 :M f56 sf 1.021 .102( and write system)J 72 120 :M -.096(\(1\) as a system of differential equations in )A 273 113 9 9 rC 282 122 :M psb currentpoint pse 273 113 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 288 3 -1 roll exch div scale currentpoint translate 64 -37 translate -3 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (u) show end pse gR gS 17 23 560 797 rC 282 120 :M f56 sf -.081( and )A 305 113 9 9 rC 314 122 :M psb currentpoint pse 305 113 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 288 3 -1 roll exch div scale currentpoint translate 64 -37 translate -3 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show end pse gR gS 17 23 560 797 rC 314 120 :M f56 sf (:)S 72 185 :M ( )S 132 142 316 80 rC 448 222 :M psb currentpoint pse 132 142 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10112 div 2560 3 -1 roll exch div scale currentpoint translate 64 49 translate 261 355 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 481 355 moveto 384 /Times-Italic f1 (u) show 819 355 moveto (t) show 934 355 moveto 384 /Times-Roman f1 (,) show 1069 355 moveto 384 /Times-Italic f1 (x) show 676 365 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1273 365 moveto (\)) show 656 903 moveto 384 /Symbol f2 (\266) show 876 903 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 238 503 moveto 1173 0 rlineto stroke 1504 602 moveto 384 /Symbol f1 (+) show 1829 355 moveto 384 /Symbol f2 (\266) show 2049 355 moveto 384 /Times-Italic f1 (v) show 2383 355 moveto (t) show 2498 355 moveto 384 /Times-Roman f1 (,) show 2633 355 moveto 384 /Times-Italic f1 (x) show 2240 365 moveto 384 1000 1171 /Symbol f3 (\() show 2837 365 moveto (\)) show 2184 903 moveto 384 /Symbol f2 (\266) show 2404 903 moveto 384 /Times-Italic f1 (x) show 1806 503 moveto 1169 0 rlineto stroke 3036 602 moveto (u) show 3374 602 moveto (t) show 3489 602 moveto 384 /Times-Roman f1 (,) show 3624 602 moveto 384 /Times-Italic f1 (x) show 3231 612 moveto 384 1000 1171 /Symbol f3 (\() show 3828 612 moveto (\)) show 4027 602 moveto 384 /Symbol f1 (+) show 4352 355 moveto 384 /Symbol f2 (\266) show 4572 355 moveto 384 /Times-Italic f1 (u) show 4910 355 moveto (t) show 5025 355 moveto 384 /Times-Roman f1 (,) show 5160 355 moveto 384 /Times-Italic f1 (x) show 4767 365 moveto 384 1000 1171 /Symbol f3 (\() show 5364 365 moveto (\)) show 4709 903 moveto 384 /Symbol f2 (\266) show 4929 903 moveto 384 /Times-Italic f1 (x) show 4329 503 moveto 1173 0 rlineto stroke 5563 602 moveto (v) show 5897 602 moveto (t) show 6012 602 moveto 384 /Times-Roman f1 (,) show 6147 602 moveto 384 /Times-Italic f1 (x) show 5754 612 moveto 384 1000 1171 /Symbol f3 (\() show 6351 612 moveto (\)) show 6550 602 moveto 384 /Symbol f1 (+) show 6873 355 moveto 384 /Symbol f2 (s) show 7162 184 moveto 224 /Times-Roman f1 (2) show 6996 903 moveto 384 ns (2) show 6852 503 moveto 480 0 rlineto stroke 7419 355 moveto 384 /Symbol f2 (\266) show 7671 184 moveto 224 /Times-Roman f1 (2) show 7806 355 moveto 384 /Times-Italic f1 (v) show 8140 355 moveto (t) show 8255 355 moveto 384 /Times-Roman f1 (,) show 8390 355 moveto 384 /Times-Italic f1 (x) show 7997 365 moveto 384 1000 1171 /Symbol f3 (\() show 8594 365 moveto (\)) show 7776 903 moveto 384 /Symbol f2 (\266) show 7996 903 moveto 384 /Times-Italic f1 (x) show 8205 732 moveto 224 /Times-Roman f1 (2) show 7396 503 moveto 1336 0 rlineto stroke 8856 602 moveto 384 /Symbol f1 (=) show 9181 602 moveto 384 /Times-Roman f1 (0) show 9395 602 moveto 384 /Times-Roman f1 ( ) show 218 1869 moveto 384 /Symbol f2 (\266) show 438 1869 moveto 384 /Times-Italic f1 (v) show 772 1869 moveto (t) show 887 1869 moveto 384 /Times-Roman f1 (,) show 1022 1869 moveto 384 /Times-Italic f1 (x) show 629 1879 moveto 384 1000 1171 /Symbol f3 (\() show 1226 1879 moveto (\)) show 611 2417 moveto 384 /Symbol f2 (\266) show 831 2417 moveto 384 /Times-Italic f1 (t) show 195 2017 moveto 1169 0 rlineto stroke 1457 2116 moveto 384 /Symbol f1 (+) show 1782 1869 moveto 384 /Symbol f2 (\266) show 2002 1869 moveto 384 /Times-Italic f1 (v) show 2336 1869 moveto (t) show 2451 1869 moveto 384 /Times-Roman f1 (,) show 2586 1869 moveto 384 /Times-Italic f1 (x) show 2193 1879 moveto 384 1000 1171 /Symbol f3 (\() show 2790 1879 moveto (\)) show 2137 2417 moveto 384 /Symbol f2 (\266) show 2357 2417 moveto 384 /Times-Italic f1 (x) show 1759 2017 moveto 1169 0 rlineto stroke 2989 2116 moveto (v) show 3323 2116 moveto (t) show 3438 2116 moveto 384 /Times-Roman f1 (,) show 3573 2116 moveto 384 /Times-Italic f1 (x) show 3180 2126 moveto 384 1000 1171 /Symbol f3 (\() show 3777 2126 moveto (\)) show 3975 2116 moveto 384 /Symbol f1 (-) show 4299 1869 moveto 384 /Symbol f2 (\266) show 4519 1869 moveto 384 /Times-Italic f1 (u) show 4857 1869 moveto (t) show 4972 1869 moveto 384 /Times-Roman f1 (,) show 5107 1869 moveto 384 /Times-Italic f1 (x) show 4714 1879 moveto 384 1000 1171 /Symbol f3 (\() show 5311 1879 moveto (\)) show 4656 2417 moveto 384 /Symbol f2 (\266) show 4876 2417 moveto 384 /Times-Italic f1 (x) show 4276 2017 moveto 1173 0 rlineto stroke 5510 2116 moveto (u) show 5848 2116 moveto (t) show 5963 2116 moveto 384 /Times-Roman f1 (,) show 6098 2116 moveto 384 /Times-Italic f1 (x) show 5705 2126 moveto 384 1000 1171 /Symbol f3 (\() show 6302 2126 moveto (\)) show 6500 2116 moveto 384 /Symbol f1 (-) show 6822 1869 moveto 384 /Symbol f2 (s) show 7111 1698 moveto 224 /Times-Roman f1 (2) show 6945 2417 moveto 384 ns (2) show 6801 2017 moveto 480 0 rlineto stroke 7368 1869 moveto 384 /Symbol f2 (\266) show 7620 1698 moveto 224 /Times-Roman f1 (2) show 7755 1869 moveto 384 /Times-Italic f1 (u) show 8093 1869 moveto (t) show 8208 1869 moveto 384 /Times-Roman f1 (,) show 8343 1869 moveto 384 /Times-Italic f1 (x) show 7950 1879 moveto 384 1000 1171 /Symbol f3 (\() show 8547 1879 moveto (\)) show 7727 2417 moveto 384 /Symbol f2 (\266) show 7947 2417 moveto 384 /Times-Italic f1 (x) show 8156 2246 moveto 224 /Times-Roman f1 (2) show 7345 2017 moveto 1340 0 rlineto stroke 8809 2116 moveto 384 /Symbol f1 (=) show 9129 2116 moveto 384 /Times-Italic f1 (a) show 9453 2116 moveto (t) show 9568 2116 moveto 384 /Times-Roman f1 (,) show 9703 2116 moveto 384 /Times-Italic f1 (x) show 9310 2126 moveto 384 1000 1171 /Symbol f3 (\() show 9907 2126 moveto (\)) show -5 355 moveto 384 /Symbol f1 (\354) show -5 1392 moveto (\355) show -5 726 moveto (\357) show -5 1099 moveto (\357) show -5 2430 moveto (\356) show -5 1766 moveto (\357) show -5 2139 moveto (\357) show end pse gR gS 17 23 560 797 rC 448 185 :M f56 sf ( \(2\))S 72 255 :M -.087(where, in the case at hand:)A 72 278 :M ( )S 216 259 169 36 rC 385 295 :M psb currentpoint pse 216 259 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5408 div 1152 3 -1 roll exch div scale currentpoint translate 64 46 translate -14 562 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (a) show 310 562 moveto (t) show 425 562 moveto 384 /Times-Roman f1 (,) show 560 562 moveto 384 /Times-Italic f1 (x) show 167 572 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 764 572 moveto (\)) show 978 562 moveto 384 /Symbol f1 (=) show 1282 562 moveto 384 /Times-Italic f1 (a) show 1478 658 moveto 224 /Times-Roman f1 (0) show 1771 562 moveto 384 /Times-Italic f1 (t) show 1886 562 moveto 384 /Times-Roman f1 (,) show 2021 562 moveto 384 /Times-Italic f1 (x) show 1628 572 moveto 384 1000 1171 /Symbol f3 (\() show 2225 572 moveto (\)) show 2439 562 moveto 384 /Symbol f1 (=) show 2753 562 moveto (-) show 3742 323 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 4035 152 moveto 224 /Times-Roman f1 (4) show 4242 323 moveto 384 /Symbol f1 (\327) show 4396 323 moveto 384 /Times-Italic f1 (x) show 3043 914 moveto 384 /Times-Roman f1 (2) show 3295 914 moveto 384 /Symbol f1 (\327) show 3567 914 moveto 384 /Symbol f2 (a) show 3936 914 moveto 384 /Symbol f1 (+) show 4227 914 moveto 384 /Symbol f2 (s) show 4516 743 moveto 224 /Times-Roman f1 (2) show 4724 914 moveto 384 /Symbol f1 (\327) show 4880 914 moveto 384 /Times-Italic f1 (t) show 3435 958 moveto 384 1000 1544 /Symbol f3 (\() show 5008 958 moveto (\)) show 5140 655 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3022 463 moveto 2288 0 rlineto stroke end pse gR gS 17 23 560 797 rC 385 278 :M f56 sf ( \(3\))S 72 310 :M -.042(As it is well known [1], setting)A 249 314 92 29 rC 341 343 :M psb currentpoint pse 249 314 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2944 div 928 3 -1 roll exch div scale currentpoint translate 64 45 translate -14 531 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (a) show 310 531 moveto (t) show 425 531 moveto 384 /Times-Roman f1 (,) show 560 531 moveto 384 /Times-Italic f1 (x) show 167 541 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 764 541 moveto (\)) show 978 531 moveto 384 /Symbol f1 (=) show 1292 531 moveto (-) show 1584 292 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 1804 292 moveto 384 /Times-Italic f1 (V) show 2228 292 moveto (t) show 2343 292 moveto 384 /Times-Roman f1 (,) show 2478 292 moveto 384 /Times-Italic f1 (x) show 2085 302 moveto 384 1000 1171 /Symbol f3 (\() show 2682 302 moveto (\)) show 1984 824 moveto 384 /Symbol f2 (\266) show 2204 824 moveto 384 /Times-Italic f1 (x) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1561 432 moveto 1259 0 rlineto stroke end pse gR gS 17 23 560 797 rC 341 332 :M f56 sf (,)S 253 362 84 29 rC 337 391 :M psb currentpoint pse 253 362 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2688 div 928 3 -1 roll exch div scale currentpoint translate 64 45 translate -3 531 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (u) show 335 531 moveto (t) show 450 531 moveto 384 /Times-Roman f1 (,) show 585 531 moveto 384 /Times-Italic f1 (x) show 192 541 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 789 541 moveto (\)) show 1003 531 moveto 384 /Symbol f1 (=) show 1344 292 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 1564 292 moveto 384 /Times-Italic f1 (R) show 1970 292 moveto (t) show 2085 292 moveto 384 /Times-Roman f1 (,) show 2220 292 moveto 384 /Times-Italic f1 (x) show 1827 302 moveto 384 1000 1171 /Symbol f3 (\() show 2424 302 moveto (\)) show 1735 824 moveto 384 /Symbol f2 (\266) show 1955 824 moveto 384 /Times-Italic f1 (x) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1321 432 moveto 1241 0 rlineto stroke end pse gR gS 17 23 560 797 rC 337 380 :M f56 sf (,)S 253 428 :M ( )S 256 410 81 29 rC 337 439 :M psb currentpoint pse 256 410 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2592 div 928 3 -1 roll exch div scale currentpoint translate 64 45 translate -3 531 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 331 531 moveto (t) show 446 531 moveto 384 /Times-Roman f1 (,) show 581 531 moveto 384 /Times-Italic f1 (x) show 188 541 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 785 541 moveto (\)) show 999 531 moveto 384 /Symbol f1 (=) show 1340 292 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 1560 292 moveto 384 /Times-Italic f1 (S) show 1903 292 moveto (t) show 2018 292 moveto 384 /Times-Roman f1 (,) show 2153 292 moveto 384 /Times-Italic f1 (x) show 1760 302 moveto 384 1000 1171 /Symbol f3 (\() show 2357 302 moveto (\)) show 1700 824 moveto 384 /Symbol f2 (\266) show 1920 824 moveto 384 /Times-Italic f1 (x) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1317 432 moveto 1178 0 rlineto stroke end pse gR gS 17 23 560 797 rC 337 428 :M f56 sf (,)S 72 454 :M -.099(system \(2\) transforms into the )A f78 sf 12 u78 :p 29.678 :m -.091(linear )A f56 sf -.124(equation)A 72 495 :M ( )S 213 476 174 30 rC 387 506 :M psb currentpoint pse 213 476 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5568 div 960 3 -1 roll exch div scale currentpoint translate 64 46 translate -8 562 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (i) show 161 562 moveto 384 /Symbol f1 (\327) show 310 562 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 599 391 moveto 224 /Times-Roman f1 (2) show 807 562 moveto 384 /Symbol f1 (\327) show 990 323 moveto 384 /Symbol f2 (\266) show 1210 323 moveto (j) show 1067 855 moveto (\266) show 1287 855 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 967 463 moveto 537 0 rlineto stroke 1612 562 moveto 384 /Symbol f1 (=) show 1926 562 moveto (-) show 2216 323 moveto 384 /Symbol f2 (s) show 2509 152 moveto 224 /Times-Roman f1 (4) show 2340 855 moveto 384 ns (2) show 2195 463 moveto 483 0 rlineto stroke 2748 562 moveto 384 /Symbol f1 (\327) show 2931 323 moveto 384 /Symbol f2 (\266) show 3183 152 moveto 224 /Times-Roman f1 (2) show 3308 323 moveto 384 /Symbol f2 (j) show 2967 855 moveto (\266) show 3187 855 moveto 384 /Times-Italic f1 (x) show 3396 684 moveto 224 /Times-Roman f1 (2) show 2908 463 moveto 694 0 rlineto stroke 3695 562 moveto 384 /Symbol f1 (+) show 3990 562 moveto 384 /Times-Italic f1 (V) show 4414 562 moveto (t) show 4529 562 moveto 384 /Times-Roman f1 (,) show 4664 562 moveto 384 /Times-Italic f1 (x) show 4271 572 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 4868 572 moveto (\)) show 5044 562 moveto 384 /Symbol f1 (\327) show 5191 562 moveto 384 /Symbol f2 (j) show end pse gR gS 17 23 560 797 rC 387 495 :M f56 sf ( \(4\))S 72 538 :M -.071(in the unknown function )A 191 526 100 16 rC 291 542 :M psb currentpoint pse 191 526 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3200 div 512 3 -1 roll exch div scale currentpoint translate 64 36 translate -13 348 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 348 moveto 384 /Times-Italic f1 (t) show 519 348 moveto 384 /Times-Roman f1 (,) show 654 348 moveto 384 /Times-Italic f1 (x) show 261 358 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 858 358 moveto (\)) show 1072 348 moveto 384 /Symbol f1 (=) show 1382 348 moveto 384 /Times-Italic f1 (e) show 1573 169 moveto 224 ns (R) show 1837 169 moveto (t) show 1917 169 moveto 224 /Times-Roman f1 (,) show 1991 169 moveto 224 /Times-Italic f1 (x) show 1740 176 moveto 224 1000 1174 /Symbol f3 (\() show 2124 176 moveto (\)) show 2216 169 moveto 224 /Symbol f1 (+) show 2364 169 moveto 224 /Times-Italic f1 (i) show 2439 169 moveto 224 /Symbol f1 (\327) show 2499 169 moveto 224 /Times-Italic f1 (S) show 2726 169 moveto (t) show 2806 169 moveto 224 /Times-Roman f1 (,) show 2880 169 moveto 224 /Times-Italic f1 (x) show 2629 176 moveto 224 1000 1174 /Symbol f3 (\() show 3013 176 moveto (\)) show end pse gR gS 17 23 560 797 rC 291 538 :M f56 sf .08 .008( . In particular, problem \(1\) reduces to the linear)J 72 557 :M -.189(equation)A 72 580 :M ( )S 183 561 215 36 rC 398 597 :M psb currentpoint pse 183 561 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6880 div 1152 3 -1 roll exch div scale currentpoint translate 64 46 translate -8 562 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (i) show 161 562 moveto 384 /Symbol f1 (\327) show 310 562 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 599 391 moveto 224 /Times-Roman f1 (2) show 807 562 moveto 384 /Symbol f1 (\327) show 990 323 moveto 384 /Symbol f2 (\266) show 1210 323 moveto (j) show 1067 855 moveto (\266) show 1287 855 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 967 463 moveto 537 0 rlineto stroke 1612 562 moveto 384 /Symbol f1 (=) show 1926 562 moveto (-) show 2216 323 moveto 384 /Symbol f2 (s) show 2509 152 moveto 224 /Times-Roman f1 (4) show 2340 855 moveto 384 ns (2) show 2195 463 moveto 483 0 rlineto stroke 2748 562 moveto 384 /Symbol f1 (\327) show 2931 323 moveto 384 /Symbol f2 (\266) show 3183 152 moveto 224 /Times-Roman f1 (2) show 3308 323 moveto 384 /Symbol f2 (j) show 2967 855 moveto (\266) show 3187 855 moveto 384 /Times-Italic f1 (x) show 3396 684 moveto 224 /Times-Roman f1 (2) show 2908 463 moveto 694 0 rlineto stroke 3695 562 moveto 384 /Symbol f1 (+) show 4638 323 moveto 384 /Symbol f2 (s) show 4931 152 moveto 224 /Times-Roman f1 (4) show 5138 323 moveto 384 /Symbol f1 (\327) show 5292 323 moveto 384 /Times-Italic f1 (x) show 5501 152 moveto 224 /Times-Roman f1 (2) show 4025 914 moveto 384 ns (4) show 4276 914 moveto 384 /Symbol f1 (\327) show 4548 914 moveto 384 /Symbol f2 (a) show 4917 914 moveto 384 /Symbol f1 (+) show 5208 914 moveto 384 /Symbol f2 (s) show 5497 743 moveto 224 /Times-Roman f1 (2) show 5705 914 moveto 384 /Symbol f1 (\327) show 5861 914 moveto 384 /Times-Italic f1 (t) show 4416 958 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1544 /Symbol f3 (\() show 5989 958 moveto (\)) show 6121 655 moveto 224 /Times-Roman f1 (2) show 3997 463 moveto 2294 0 rlineto stroke 6361 562 moveto 384 /Symbol f1 (\327) show 6508 562 moveto 384 /Symbol f2 (j) show end pse gR gS 17 23 560 797 rC 398 580 :M f56 sf ( \(5\))S 72 610 :M .208 .021(The Brownian motion )J 182 599 26 16 rC 208 615 :M psb currentpoint pse 182 599 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 209 388 moveto 224 /Times-Roman f1 (0) show 502 292 moveto 384 /Times-Italic f1 (t) show 359 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 630 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 208 610 :M f56 sf .499 .05(, as already pointed out in [2], appears, therefore, as a particular)J 72 630 :M 1.198 .12(solution of \(5\). Other solutions of \(5\) have, in fact, been found in Step 1: they are given)J 72 648 :M -.204(explicitly by:)A 126 719 :M ( )S 132 670 334 92 rC 466 762 :M psb currentpoint pse 132 670 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10688 div 2944 3 -1 roll exch div scale currentpoint translate 64 45 translate -13 793 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 259 889 moveto 224 /Times-Italic f1 (n) show 555 793 moveto 384 ns (t) show 670 793 moveto 384 /Times-Roman f1 (,) show 805 793 moveto 384 /Times-Italic f1 (x) show 412 803 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1009 803 moveto (\)) show 1223 793 moveto 384 /Symbol f1 (=) show 3049 554 moveto 384 /Times-Roman f1 (1) show 1689 1176 moveto (2) show 1941 1176 moveto 384 /Symbol f1 (\327) show 2098 1176 moveto 384 /Symbol f2 (p) show 2408 1176 moveto 384 /Symbol f1 (\327) show 2680 1176 moveto 384 /Symbol f2 (a) show 3049 1176 moveto 384 /Symbol f1 (+) show 3340 1176 moveto 384 /Symbol f2 (s) show 3629 1005 moveto 224 /Times-Roman f1 (2) show 3837 1176 moveto 384 /Symbol f1 (\327) show 3993 1176 moveto 384 /Times-Italic f1 (t) show 2548 1220 moveto 384 1000 1544 /Symbol f3 (\() show 4121 1220 moveto (\)) show 1553 1237 moveto 384 1000 1731 /Symbol f3 (\() show 4248 1237 moveto (\)) show 4362 886 moveto 224 /Times-Roman f1 (1) show 4484 886 moveto 224 /Times-Roman f1 (/) show 4581 886 moveto 224 /Times-Roman f1 (4) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1541 694 moveto 3209 0 rlineto stroke 4820 793 moveto 384 /Symbol f1 (\327) show 5656 554 moveto 384 /Times-Roman f1 (1) show 5128 1145 moveto 384 /Times-Roman f1 (2) show 5340 974 moveto 224 /Times-Italic f1 (n) show 5551 1145 moveto 384 /Symbol f1 (\327) show 5708 1145 moveto 384 /Times-Italic f1 (n) show 5919 1145 moveto 384 /Times-Roman f1 (!) show 4992 1189 moveto 384 1000 1544 /Symbol f3 (\() show 6026 1189 moveto (\)) show 6140 886 moveto 224 /Times-Roman f1 (1) show 6262 886 moveto 224 /Times-Roman f1 (/) show 6355 886 moveto 224 /Times-Roman f1 (2) show 4980 694 moveto 1545 0 rlineto stroke 6595 793 moveto 384 /Symbol f1 (\327) show 6748 793 moveto 384 /Times-Italic f1 (H) show 7047 889 moveto 224 ns (n) show 8519 554 moveto 384 ns (x) show 7687 1168 moveto 384 /Times-Roman f1 (2) show 7939 1168 moveto 384 /Symbol f1 (\327) show 8211 1168 moveto 384 /Symbol f2 (a) show 8580 1168 moveto 384 /Symbol f1 (+) show 8871 1168 moveto 384 /Symbol f2 (s) show 9160 997 moveto 224 /Times-Roman f1 (2) show 9368 1168 moveto 384 /Symbol f1 (\327) show 9524 1168 moveto 384 /Times-Italic f1 (t) show 8079 1212 moveto 384 1000 1544 /Symbol f3 (\() show 9652 1212 moveto (\)) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 2354 622 384 7436 1356 16 sqr 7404 694 moveto 2418 0 rlineto stroke 7205 355 moveto 384 /Symbol f1 (\346) show 7205 1276 moveto (\350) show 7205 806 moveto (\347) show 7205 1057 moveto (\347) show 9834 355 moveto (\366) show 9834 1276 moveto (\370) show 9834 806 moveto (\367) show 9834 1057 moveto (\367) show 10076 793 moveto (\327) show -26 2333 moveto (\327) show 94 2333 moveto 384 /Times-Roman f1 (exp) show 848 2333 moveto 384 /Symbol f1 (-) show 2006 2094 moveto 384 /Times-Italic f1 (x) show 2215 1923 moveto 224 /Times-Roman f1 (2) show 1145 2628 moveto 384 ns (4) show 1396 2628 moveto 384 /Symbol f1 (\327) show 1668 2628 moveto 384 /Symbol f2 (a) show 2037 2628 moveto 384 /Symbol f1 (+) show 2328 2628 moveto 384 /Symbol f2 (s) show 2617 2457 moveto 224 /Times-Roman f1 (2) show 2825 2628 moveto 384 /Symbol f1 (\327) show 2981 2628 moveto 384 /Times-Italic f1 (t) show 1536 2672 moveto 384 1000 1544 /Symbol f3 (\() show 3109 2672 moveto (\)) show 1117 2234 moveto 2130 0 rlineto stroke 3340 2333 moveto 384 /Symbol f1 (+) show 3634 2333 moveto 384 /Times-Italic f1 (i) show 3803 2333 moveto 384 /Symbol f1 (\327) show 5036 2094 moveto 384 /Times-Italic f1 (x) show 5245 1923 moveto 224 /Times-Roman f1 (2) show 4175 2628 moveto 384 ns (4) show 4426 2628 moveto 384 /Symbol f1 (\327) show 4698 2628 moveto 384 /Symbol f2 (a) show 5067 2628 moveto 384 /Symbol f1 (+) show 5358 2628 moveto 384 /Symbol f2 (s) show 5647 2457 moveto 224 /Times-Roman f1 (2) show 5855 2628 moveto 384 /Symbol f1 (\327) show 6011 2628 moveto 384 /Times-Italic f1 (t) show 4566 2672 moveto 384 1000 1544 /Symbol f3 (\() show 6139 2672 moveto (\)) show 4147 2234 moveto 2130 0 rlineto stroke 6369 2333 moveto 384 /Symbol f1 (-) show 6660 2094 moveto 384 /Times-Roman f1 (1) show 6904 2094 moveto 384 /Symbol f1 (+) show 7195 2094 moveto 384 /Times-Roman f1 (2) show 7447 2094 moveto 384 /Symbol f1 (\327) show 7604 2094 moveto 384 /Times-Italic f1 (n) show 7155 2626 moveto 384 /Times-Roman f1 (4) show 6670 2234 moveto 1154 0 rlineto stroke 7894 2333 moveto 384 /Symbol f1 (\327) show 8047 2333 moveto 384 /Times-Roman f1 (ln) show 8574 2094 moveto 384 /Symbol f2 (a) show 8943 2094 moveto 384 /Symbol f1 (+) show 9234 2094 moveto 384 /Symbol f2 (s) show 9523 1923 moveto 224 /Times-Roman f1 (2) show 9731 2094 moveto 384 /Symbol f1 (\327) show 9887 2094 moveto 384 /Times-Italic f1 (t) show 9146 2626 moveto 384 /Symbol f2 (a) show 8557 2234 moveto 1469 0 rlineto stroke 8358 2094 moveto 384 /Symbol f1 (\346) show 8358 2617 moveto (\350) show 8358 2398 moveto (\347) show 10038 2094 moveto (\366) show 10038 2617 moveto (\370) show 10038 2398 moveto (\367) show 3948 2007 moveto (\346) show 3948 2704 moveto (\350) show 3948 2458 moveto (\347) show 10222 2007 moveto (\366) show 10222 2704 moveto (\370) show 10222 2458 moveto (\367) show 653 1975 moveto (\346) show 653 2736 moveto (\350) show 653 2426 moveto (\347) show 10406 1975 moveto (\366) show 10406 2736 moveto (\370) show 10406 2426 moveto (\367) show end pse endp showpage %%Page: 5 5 %%BeginPageSetup initializepage (defalco; pag: 5 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (5)S 70 97 :M -.069(Step 3:)A f103 sf ( )S f56 sf -.071(The correspondence set in the previous step between solutions )A 404 86 21 15 rC 425 101 :M psb currentpoint pse 404 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 292 moveto (t) show 194 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 425 97 :M f56 sf .226 .023( of \(1\) and solutions)J 70 104 34 15 rC 104 119 :M psb currentpoint pse 70 104 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -13 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 292 moveto 384 /Times-Italic f1 (t) show 519 292 moveto 384 /Times-Roman f1 (,) show 654 292 moveto 384 /Times-Italic f1 (x) show 261 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 858 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 104 115 :M f56 sf 1.818 .182( of \(5\) requires in particular that )J 280 104 21 15 rC 301 119 :M psb currentpoint pse 280 104 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 292 moveto (t) show 194 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 301 115 :M f56 sf 2.005 .2( have probability density )J 436 102 83 18 rC 519 120 :M psb currentpoint pse 436 102 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2656 div 576 3 -1 roll exch div scale currentpoint translate 64 34 translate -19 382 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (r) show 361 382 moveto 384 /Times-Italic f1 (t) show 476 382 moveto 384 /Times-Roman f1 (,) show 611 382 moveto 384 /Times-Italic f1 (x) show 218 392 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 815 392 moveto (\)) show 1029 382 moveto 384 /Symbol f1 (=) show 1382 382 moveto 384 /Symbol f2 (j) show 1799 382 moveto 384 /Times-Italic f1 (t) show 1914 382 moveto 384 /Times-Roman f1 (,) show 2049 382 moveto 384 /Times-Italic f1 (x) show 1656 392 moveto 384 1000 1171 /Symbol f3 (\() show 2253 392 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1355 57 moveto 0 452 rlineto stroke 2399 57 moveto 0 452 rlineto stroke 2433 152 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 519 115 :M f56 sf (.)S 70 135 :M .503 .05(Because of the )J f78 sf 12 u78 :p 41.211 :m .137(linearity)A f56 sf .405 .04( of equation \(5\) and of )J f78 sf 12 u78 :p 221.874 :m .817 .082(orthonormality and completeness of Hermite)J 70 153 :M 44.572 :m -.009(functions)A f56 sf -.009(, every solution of problem \(1\) corresponds to a normalised linear combination of the)A 70 170 :M .289 .029(above solutions )J 149 159 38 15 rC 187 174 :M psb currentpoint pse 149 159 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 480 3 -1 roll exch div scale currentpoint translate 64 -441 translate -13 793 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 259 889 moveto 224 /Times-Italic f1 (n) show 555 793 moveto 384 ns (t) show 670 793 moveto 384 /Times-Roman f1 (,) show 805 793 moveto 384 /Times-Italic f1 (x) show 412 803 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1009 803 moveto (\)) show end pse gR gS 17 23 560 797 rC 187 170 :M f56 sf .553 .055( \(with coefficients determined, for instance, by the )J f78 sf 12 u78 :p 83.422 :m .9 .09(initial conditions)J 70 177 34 15 rC 104 192 :M psb currentpoint pse 70 177 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 480 3 -1 roll exch div scale currentpoint translate 64 -6173 translate -3 6525 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 326 6525 moveto 384 /Times-Roman f1 (0) show 516 6525 moveto 384 /Times-Roman f1 (,) show 651 6525 moveto 384 /Times-Italic f1 (x) show 188 6535 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 855 6535 moveto (\)) show end pse gR gS 17 23 560 797 rC 104 188 :M f56 sf -.081( and )A 127 177 34 15 rC 161 192 :M psb currentpoint pse 127 177 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -19 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (r) show 356 292 moveto 384 /Times-Roman f1 (0) show 546 292 moveto 384 /Times-Roman f1 (,) show 681 292 moveto 384 /Times-Italic f1 (x) show 218 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 885 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 161 188 :M f56 sf -.108( imposed on the current velocity field and on the probability density field\).)A 70 225 :M 1.042 .104(The paradigmatic example proposed in [2], the presentation of which has been given -and)J 70 243 :M -.057(somewhat enriched- above, helps set the stage of the problem we wish to address here.)A 70 279 :M -.086(As we have explicitly observed in our particular example, the solution of the problem of finding)A 70 297 :M 2.07 .207(the diffusion processes having an assigned diffusion coefficient and an assigned mean)J 70 321 :M .786 .079(acceleration, written here as )J 213 301 117 31 rC 330 332 :M psb currentpoint pse 213 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3744 div 992 3 -1 roll exch div scale currentpoint translate -1234 -1778 translate 1284 2418 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (a) show 1608 2418 moveto (t) show 1723 2418 moveto 384 /Times-Roman f1 (,) show 1858 2418 moveto 384 /Times-Italic f1 (x) show 1465 2428 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 2062 2428 moveto (\)) show 2276 2418 moveto 384 /Symbol f1 (=) show 2590 2418 moveto (-) show 2939 2179 moveto 384 /Times-Roman f1 (1) show 2887 2711 moveto 384 /Times-Italic f1 (m) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2859 2319 moveto 353 0 rlineto stroke 3299 2152 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 3519 2152 moveto 384 /Times-Italic f1 (V) show 3943 2152 moveto (t) show 4058 2152 moveto 384 /Times-Roman f1 (,) show 4190 2152 moveto 384 /Times-Italic f1 (q) show 4536 2152 moveto (t) show 4393 2162 moveto 384 1000 1171 /Symbol f3 (\() show 4664 2162 moveto (\)) show 3800 2180 moveto 384 1000 1364 /Symbol f3 (\() show 4791 2180 moveto (\)) show 3896 2711 moveto 384 /Symbol f2 (\266) show 4116 2711 moveto 384 /Times-Italic f1 (x) show 3276 2319 moveto 1653 0 rlineto stroke end pse gR gS 17 23 560 797 rC 330 321 :M f56 sf 1.754 .175( , is equivalent to the solution of the)J 70 347 :M -.165(problem)A 184 351 222 33 rC 406 384 :M psb currentpoint pse 184 351 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7104 div 1056 3 -1 roll exch div scale currentpoint translate 64 -3331 translate -8 4035 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (i) show 161 4035 moveto 384 /Symbol f1 (\327) show 444 4035 moveto 384 /Times-Italic f1 (m) show 807 4035 moveto 384 /Symbol f1 (\327) show 956 4035 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 1245 3864 moveto 224 /Times-Roman f1 (2) show 301 4079 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1544 /Symbol f3 (\() show 1404 4079 moveto (\)) show 1580 4035 moveto 384 /Symbol f1 (\327) show 1763 3796 moveto 384 /Symbol f2 (\266) show 1983 3796 moveto (j) show 1840 4328 moveto (\266) show 2060 4328 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1740 3936 moveto 537 0 rlineto stroke 2385 4035 moveto 384 /Symbol f1 (=) show 2699 4035 moveto (-) show 3123 3740 moveto 384 /Times-Italic f1 (m) show 3486 3740 moveto 384 /Symbol f1 (\327) show 3635 3740 moveto 384 /Symbol f2 (s) show 3928 3569 moveto 224 /Times-Roman f1 (4) show 2980 3784 moveto 384 1000 1544 /Symbol f3 (\() show 4086 3784 moveto (\)) show 3500 4328 moveto 384 /Times-Roman f1 (2) show 2968 3936 moveto 1256 0 rlineto stroke 4294 4035 moveto 384 /Symbol f1 (\327) show 4477 3796 moveto 384 /Symbol f2 (\266) show 4729 3625 moveto 224 /Times-Roman f1 (2) show 4854 3796 moveto 384 /Symbol f2 (j) show 4513 4328 moveto (\266) show 4733 4328 moveto 384 /Times-Italic f1 (x) show 4942 4157 moveto 224 /Times-Roman f1 (2) show 4454 3936 moveto 694 0 rlineto stroke 5241 4035 moveto 384 /Symbol f1 (+) show 5536 4035 moveto 384 /Times-Italic f1 (V) show 5960 4035 moveto (t) show 6075 4035 moveto 384 /Times-Roman f1 (,) show 6210 4035 moveto 384 /Times-Italic f1 (x) show 5817 4045 moveto 384 1000 1171 /Symbol f3 (\() show 6414 4045 moveto (\)) show 6590 4035 moveto 384 /Symbol f1 (\327) show 6737 4035 moveto 384 /Symbol f2 (j) show end pse gR gS 17 23 560 797 rC 406 373 :M f56 sf (;)S 70 397 :M .679 .068(it is, therefore, a fact that the )J f78 sf 12 u78 :p 67.874 :m .27(mathematical)A f56 sf .763 .076( problem of solving in )J 401 385 32 17 rC 401 402 :M f133 sf ( )S 433 402 :M psb currentpoint pse 401 385 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1024 div 544 3 -1 roll exch div scale currentpoint translate 64 -515 translate -4 899 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (L) show 203 728 moveto 224 ns (2) show 468 899 moveto 384 /Math f1 (R) show 353 924 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1333 /Symbol f3 (\() show 809 924 moveto (\)) show end pse gR gS 17 23 560 797 rC 433 397 :M f56 sf 1.673 .167( the Schr\232dinger)J 70 417 :M -.189(equation)A 209 421 175 30 rC 209 451 :M f167 sf .269 .027( )J 384 451 :M psb currentpoint pse 209 421 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5600 div 960 3 -1 roll exch div scale currentpoint translate 64 -530 translate -8 1138 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (i) show 161 1138 moveto 384 /Symbol f1 (\327) show 313 1138 moveto 384 /MT-Extra f1 (h) show 564 1138 moveto 384 /Symbol f1 (\327) show 747 899 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 967 899 moveto (j) show 824 1431 moveto (\266) show 1044 1431 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 724 1039 moveto 537 0 rlineto stroke 1369 1138 moveto 384 /Symbol f1 (=) show 1683 1138 moveto (-) show 2153 899 moveto 384 /MT-Extra f1 (h) show 2360 728 moveto 224 /Times-Roman f1 (2) show 1973 1431 moveto 384 ns (2) show 2225 1431 moveto 384 /Symbol f1 (\327) show 2381 1431 moveto 384 /Times-Italic f1 (m) show 1952 1039 moveto 754 0 rlineto stroke 2776 1138 moveto 384 /Symbol f1 (\327) show 2959 899 moveto 384 /Symbol f2 (\266) show 3211 728 moveto 224 /Times-Roman f1 (2) show 3336 899 moveto 384 /Symbol f2 (j) show 2995 1431 moveto (\266) show 3215 1431 moveto 384 /Times-Italic f1 (x) show 3424 1260 moveto 224 /Times-Roman f1 (2) show 2936 1039 moveto 694 0 rlineto stroke 3723 1138 moveto 384 /Symbol f1 (+) show 4018 1138 moveto 384 /Times-Italic f1 (V) show 4442 1138 moveto (t) show 4557 1138 moveto 384 /Times-Roman f1 (,) show 4692 1138 moveto 384 /Times-Italic f1 (x) show 4299 1148 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 4896 1148 moveto (\)) show 5072 1138 moveto 384 /Symbol f1 (\327) show 5219 1138 moveto 384 /Symbol f2 (j) show end pse gR gS 17 23 560 797 rC 70 466 :M f56 sf .412 .041(is equivalent to the )J f78 sf 12 u78 :p 66.366 :m .144(mathematical)A f56 sf .473 .047( problem of solving, in the class of diffusion processes, the)J 70 484 :M -.066(system)A 200 506 192 89 rC 200 595 :M f167 sf .269 .027( )J 392 595 :M psb currentpoint pse 200 506 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6144 div 2848 3 -1 roll exch div scale currentpoint translate 64 -514 translate 251 1287 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (lim) show 193 1526 moveto 224 /Times-Italic f1 (h) show 324 1526 moveto 224 /Symbol f1 (\256) show 565 1526 moveto 224 /Times-Roman f1 (0) show 702 1427 moveto 160 /Symbol f1 (+) show 880 1287 moveto 384 /Times-Italic f1 (E) show 1139 1383 moveto 224 ns (t) show 1599 1021 moveto 384 ns (q) show 1945 1021 moveto (t) show 2145 1021 moveto 384 /Symbol f1 (+) show 2444 1021 moveto 384 /Times-Italic f1 (h) show 1802 1031 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 2653 1031 moveto (\)) show 2851 1021 moveto 384 /Symbol f1 (-) show 3143 1021 moveto 384 /Times-Italic f1 (q) show 3489 1021 moveto (t) show 3346 1031 moveto 384 1000 1171 /Symbol f3 (\() show 3617 1031 moveto (\)) show 1461 1049 moveto 384 1000 1364 /Symbol f3 (\() show 3744 1049 moveto (\)) show 3876 792 moveto 224 /Times-Roman f1 (2) show 2652 1580 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1449 1188 moveto 2597 0 rlineto stroke 1250 963 moveto 384 /Symbol f1 (\346) show 1250 1656 moveto (\350) show 1250 1414 moveto (\347) show 4058 963 moveto (\366) show 4058 1656 moveto (\370) show 4058 1414 moveto (\367) show 4338 1287 moveto (=) show 4738 1048 moveto 384 /MT-Extra f1 (h) show 4684 1580 moveto 384 /Times-Italic f1 (m) show 4656 1188 moveto 353 0 rlineto stroke 5073 1287 moveto 384 /Times-Roman f1 ( ) show 196 2994 moveto 384 /Times-Italic f1 (m) show 559 2994 moveto 384 /Symbol f1 (\327) show 744 2755 moveto 384 /Times-Italic f1 (D) show 1026 2584 moveto 224 /Times-Roman f1 (*) show 1148 2755 moveto 384 /Times-Italic f1 (Dq) show 1771 2755 moveto (t) show 1628 2765 moveto 384 1000 1171 /Symbol f3 (\() show 1899 2765 moveto (\)) show 2098 2755 moveto 384 /Symbol f1 (+) show 2393 2755 moveto 384 /Times-Italic f1 (DD) show 2952 2584 moveto 224 /Times-Roman f1 (*) show 3072 2755 moveto 384 /Times-Italic f1 (q) show 3418 2755 moveto (t) show 3275 2765 moveto 384 1000 1171 /Symbol f3 (\() show 3546 2765 moveto (\)) show 2105 3287 moveto 384 /Times-Roman f1 (2) show 719 2895 moveto 2965 0 rlineto stroke 3792 2994 moveto 384 /Symbol f1 (=) show 4106 2994 moveto (-) show 4398 2728 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 4618 2728 moveto 384 /Times-Italic f1 (V) show 5042 2728 moveto (t) show 5157 2728 moveto 384 /Times-Roman f1 (,) show 5289 2728 moveto 384 /Times-Italic f1 (q) show 5635 2728 moveto (t) show 5492 2738 moveto 384 1000 1171 /Symbol f3 (\() show 5763 2738 moveto (\)) show 4899 2756 moveto 384 1000 1364 /Symbol f3 (\() show 5890 2756 moveto (\)) show 4995 3287 moveto 384 /Symbol f2 (\266) show 5215 3287 moveto 384 /Times-Italic f1 (x) show 4375 2895 moveto 1653 0 rlineto stroke -5 931 moveto 384 /Symbol f1 (\354) show -5 2115 moveto (\355) show -5 1302 moveto (\357) show -5 1675 moveto (\357) show -5 1823 moveto (\357) show -5 3300 moveto (\356) show -5 2489 moveto (\357) show -5 2862 moveto (\357) show -5 3186 moveto (\357) show end pse gR gS 17 23 560 797 rC 70 628 :M f56 sf 1.318 .132(Much harder is the )J f78 sf 12 u78 :p 43.319 :m .417(physical)A f56 sf 1.637 .164( question whether quantum mechanics can be replaced by a)J 70 646 :M .641 .064(stochastic description in terms of stochastic processes. The paradigmatic example discussed)J 70 664 :M .53 .053(above helps focusing some of the issues raised by this question; for instance, one would )J f78 sf 12 u78 :p 15.791 :m .229(not)A 70 680 :M f56 sf -.106(say that the Brownian motion )A 212 668 63 18 rC 275 686 :M psb currentpoint pse 212 668 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2016 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate 153 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 424 moveto 224 /Times-Roman f1 (0) show 664 328 moveto 384 /Times-Italic f1 (t) show 521 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 338 moveto (\)) show 906 328 moveto 384 /Times-Roman f1 (,) show 1043 328 moveto 384 /Times-Italic f1 (t) show 1251 328 moveto 384 /Symbol f1 (\263) show 1555 328 moveto 384 /Times-Roman f1 (0) show -22 361 moveto 384 1000 1401 /Symbol f3 ({) show 1739 361 moveto (}) show end pse gR gS 17 23 560 797 rC 275 680 :M f56 sf -.09( is a quantum mechanical phenomenon only because)A 70 701 :M .884 .088(it appears as a particular solution of the linear problem \(5\). The main point is \(even not to)J 70 719 :M -.005(consider the issue of the numerical value of the diffusion coefficient\) that there seems to be, in)A 70 735 :M -.106(the )A f78 sf 12 u78 :p 86.52 :m -.133(phenomenological)A f56 sf -.123( context described by the Brownian motion )A 379 724 26 16 rC 405 740 :M psb currentpoint pse 379 724 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate -98 24 translate 153 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 371 424 moveto 224 /Times-Roman f1 (0) show 664 328 moveto 384 /Times-Italic f1 (t) show 521 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 792 338 moveto (\)) show end pse gR gS 17 23 560 797 rC 405 735 :M f56 sf .059 .006(, )J f78 sf 12 u78 :p 12.143 :m .071(no)A f56 sf .192 .019( useful role for other)J 70 754 :M -.037(solutions )A 116 743 27 15 rC 143 758 :M psb currentpoint pse 116 743 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 480 3 -1 roll exch div scale currentpoint translate -98 27 translate 153 325 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 374 421 moveto 224 ns (n) show 670 325 moveto 384 ns (t) show 527 335 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 798 335 moveto (\)) show end pse gR gS 17 23 560 797 rC 143 754 :M f56 sf .14 .014( of the same problem \(5\); moreover, the fact that problem \(5\) is linear is of no)J endp showpage %%Page: 6 6 %%BeginPageSetup initializepage (defalco; pag: 6 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (6)S 70 94 :M 1.662 .166(practical relevance because there seems to be no useful application, for instance, of the)J 70 111 :M -.083(processes corresponding to linear combinations of the functions )A 374 100 38 15 rC 412 115 :M psb currentpoint pse 374 100 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 480 3 -1 roll exch div scale currentpoint translate 64 -441 translate -13 793 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 259 889 moveto 224 /Times-Italic f1 (n) show 555 793 moveto 384 ns (t) show 670 793 moveto 384 /Times-Roman f1 (,) show 805 793 moveto 384 /Times-Italic f1 (x) show 412 803 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1009 803 moveto (\)) show end pse gR gS 17 23 560 797 rC 412 111 :M f56 sf (.)S 70 148 :M .815 .081(For the same reason, the observation made in Step 1 that problem \(5\), under the additional)J 70 164 :M -.131(constraint )A 119 153 78 16 rC 197 169 :M psb currentpoint pse 119 153 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2496 div 512 3 -1 roll exch div scale currentpoint translate 64 -210 translate -3 562 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 331 562 moveto (t) show 446 562 moveto 384 /Times-Roman f1 (,) show 581 562 moveto 384 /Times-Italic f1 (x) show 188 572 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 785 572 moveto (\)) show 999 562 moveto 384 /Symbol f1 (=) show 1314 562 moveto 384 /Times-Italic f1 (v) show 1520 658 moveto 224 /Times-Roman f1 (0) show 1813 562 moveto 384 /Times-Italic f1 (t) show 1928 562 moveto 384 /Times-Roman f1 (,) show 2063 562 moveto 384 /Times-Italic f1 (x) show 1670 572 moveto 384 1000 1171 /Symbol f3 (\() show 2267 572 moveto (\)) show end pse gR gS 17 23 560 797 rC 197 164 :M f56 sf -.1(, admits a countable infinity of solutions does )A f78 sf 12 u78 :p 15.001 :m -.111(not)A f56 sf -.105( mean that there is a)A 70 184 :M 1.259 .126("quantized" observable - number of nodes in the probability density - of relevance in the)J 70 202 :M -.099(phenomenological context of Brownian motion.)A 70 238 :M .55 .055(The point is here that quantum mechanics is not only the \(unrestrictedly linear\) Schr\232dinger)J 70 254 :M -.135(theory of the evolution of the initial condition )A 289 242 39 18 rC 328 260 :M psb currentpoint pse 289 242 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1248 div 576 3 -1 roll exch div scale currentpoint translate 64 -520 translate -13 904 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 904 moveto 384 /Times-Italic f1 (t) show 538 1000 moveto 224 /Times-Roman f1 (0) show 684 904 moveto 384 /Times-Roman f1 (,) show 819 904 moveto 384 /Times-Italic f1 (x) show 261 934 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1023 934 moveto (\)) show end pse gR gS 17 23 560 797 rC 328 254 :M f56 sf -.172( at a given initial instant )A 442 244 11 15 rC 453 259 :M psb currentpoint pse 442 244 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 352 div 480 3 -1 roll exch div scale currentpoint translate -344 -8 translate 404 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 538 424 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 453 254 :M f56 sf -.134( into )A 477 243 34 15 rC 511 258 :M psb currentpoint pse 477 243 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 480 3 -1 roll exch div scale currentpoint translate 64 -516 translate -13 868 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 868 moveto 384 /Times-Italic f1 (t) show 519 868 moveto 384 /Times-Roman f1 (,) show 654 868 moveto 384 /Times-Italic f1 (x) show 261 878 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 858 878 moveto (\)) show end pse gR gS 17 23 560 797 rC 511 254 :M f56 sf .283 .028( at)J 70 273 :M -.12(any instant )A 124 265 7 10 rC 131 275 :M psb currentpoint pse 124 265 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -344 -727 translate 404 983 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 131 273 :M f56 sf -.111(, but also an interpretative scheme of the meaning of )A 380 261 39 18 rC 419 279 :M psb currentpoint pse 380 261 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1248 div 576 3 -1 roll exch div scale currentpoint translate 64 -520 translate -13 904 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 904 moveto 384 /Times-Italic f1 (t) show 538 1000 moveto 224 /Times-Roman f1 (0) show 684 904 moveto 384 /Times-Roman f1 (,) show 819 904 moveto 384 /Times-Italic f1 (x) show 261 934 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1023 934 moveto (\)) show end pse gR gS 17 23 560 797 rC 419 273 :M f56 sf -.164( at each fixed )A 484 263 11 15 rC 495 278 :M psb currentpoint pse 484 263 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 352 div 480 3 -1 roll exch div scale currentpoint translate -344 -8 translate 404 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 538 424 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 495 273 :M f56 sf (.)S 70 313 :M .749 .075(This interpretative scheme is by no means exhausted by the statement that )J 443 298 46 22 rC 489 320 :M psb currentpoint pse 443 298 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1472 div 704 3 -1 roll exch div scale currentpoint translate 64 -629 translate 35 1109 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 452 1109 moveto 384 /Times-Italic f1 (t) show 586 1205 moveto 224 /Times-Roman f1 (0) show 732 1109 moveto 384 /Times-Roman f1 (,) show 867 1109 moveto 384 /Times-Italic f1 (x) show 309 1139 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1071 1139 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 8 748 moveto 0 524 rlineto stroke 1217 748 moveto 0 524 rlineto stroke 1251 843 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 489 313 :M f56 sf 2.053 .205( is the)J 70 335 :M 1.434 .143(probability density of the position observable \(on this statement quantum mechanics and)J 70 353 :M -.103(stochastic mechanics agree, essentially by construction of the latter\).)A 70 387 :M .113 .011(Part of the interpretative scheme of )J 243 375 34 18 rC 277 393 :M psb currentpoint pse 243 375 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 576 3 -1 roll exch div scale currentpoint translate 64 -635 translate -13 1019 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 1019 moveto 384 /Times-Italic f1 (t) show 538 1115 moveto 224 /Times-Roman f1 (0) show 684 1019 moveto 384 /Times-Roman f1 (,) show 799 1019 moveto 384 /Symbol f1 (\327) show 261 1049 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 884 1049 moveto (\)) show end pse gR gS 17 23 560 797 rC 277 387 :M f56 sf .769 .077( is, for instance, also the statement, that we make)J 70 408 :M .323 .032(here in the simplest possible context of a particle moving on the real line )J f133 sf .218(R)A f56 sf .35 .035(, that the random)J 70 425 :M -.146(variable, call it )A 145 412 49 20 rC 194 432 :M psb currentpoint pse 145 412 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1568 div 640 3 -1 roll exch div scale currentpoint translate 64 -635 translate 0 1051 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 326 1051 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 743 1051 moveto 384 /Times-Italic f1 (t) show 877 1147 moveto 224 /Times-Roman f1 (0) show 1023 1051 moveto 384 /Times-Roman f1 (,) show 1138 1051 moveto 384 /Symbol f1 (\327) show 600 1081 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1223 1081 moveto (\)) show 192 1098 moveto 384 1000 1580 /Symbol f3 (\() show 1350 1098 moveto (\)) show end pse gR gS 17 23 560 797 rC 194 425 :M f56 sf -.109(, defined by having characteristic function)A 202 433 189 35 rC 391 468 :M psb currentpoint pse 202 433 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6048 div 1120 3 -1 roll exch div scale currentpoint translate 64 39 translate -4 601 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) show 246 752 moveto 224 /Times-Italic f1 (p) show 463 752 moveto 224 /Symbol f2 (j) show 734 752 moveto 224 /Times-Italic f1 (t) show 825 808 moveto 160 /Times-Roman f1 (0) show 951 752 moveto 224 /Times-Roman f1 (,) show 1013 752 moveto 224 /Symbol f1 (\327) show 637 770 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 224 1000 1383 /Symbol f3 (\() show 1076 770 moveto (\)) show 372 780 moveto 224 1000 1571 /Symbol f3 (\() show 1164 780 moveto (\)) show 1414 601 moveto 384 /Symbol f2 (t) show 1270 611 moveto 384 1000 1171 /Symbol f3 (\() show 1633 611 moveto (\)) show 1847 601 moveto 384 /Symbol f1 (=) show 2448 601 moveto 384 /Symbol f2 (j) show 2865 601 moveto 384 /Times-Italic f1 (t) show 2999 697 moveto 224 /Times-Roman f1 (0) show 3145 601 moveto 384 /Times-Roman f1 (,) show 3280 601 moveto 384 /Times-Italic f1 (x) show 2722 631 moveto 384 1000 1388 /Symbol f3 (\() show 3484 631 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2461 217 moveto 1129 0 rlineto stroke 2163 1020 moveto 224 /Symbol f1 (-) show 2309 1020 moveto (\245) show 2163 152 moveto (+) show 2310 152 moveto (\245) show 2230 735 moveto 576 ns (\362) show 3660 601 moveto 384 ns (\327) show 3807 601 moveto 384 /Symbol f2 (j) show 4224 601 moveto 384 /Times-Italic f1 (t) show 4358 697 moveto 224 /Times-Roman f1 (0) show 4504 601 moveto 384 /Times-Roman f1 (,) show 4639 601 moveto 384 /Times-Italic f1 (x) show 4915 601 moveto 384 /Symbol f1 (+) show 5214 601 moveto 384 /Symbol f2 (t) show 4081 631 moveto 384 1000 1388 /Symbol f3 (\() show 5433 631 moveto (\)) show 5562 601 moveto 384 /Times-Italic f1 (dx) show end pse gR gS 17 23 560 797 rC 70 483 :M f56 sf .339 .034(has some relevance in the description of the behaviour of the process having current velocity)J 70 499 :M -.154(field and probability density field at time )A 263 489 11 15 rC 274 504 :M psb currentpoint pse 263 489 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 352 div 480 3 -1 roll exch div scale currentpoint translate -344 -8 translate 404 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 538 424 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 274 499 :M f56 sf -.132( determined by )A 350 487 34 18 rC 384 505 :M psb currentpoint pse 350 487 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 576 3 -1 roll exch div scale currentpoint translate 64 -635 translate -13 1019 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 1019 moveto 384 /Times-Italic f1 (t) show 538 1115 moveto 224 /Times-Roman f1 (0) show 684 1019 moveto 384 /Times-Roman f1 (,) show 799 1019 moveto 384 /Symbol f1 (\327) show 261 1049 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 884 1049 moveto (\)) show end pse gR gS 17 23 560 797 rC 384 499 :M f56 sf .29 .029( and that, for "some" reason,)J 70 520 :M -.107(it deserves the name of "linear momentum".)A 70 554 :M -.107(In much the same way, part of the interpretative scheme of )A 349 542 34 18 rC 383 560 :M psb currentpoint pse 349 542 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 576 3 -1 roll exch div scale currentpoint translate 64 -635 translate -13 1019 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 1019 moveto 384 /Times-Italic f1 (t) show 538 1115 moveto 224 /Times-Roman f1 (0) show 684 1019 moveto 384 /Times-Roman f1 (,) show 799 1019 moveto 384 /Symbol f1 (\327) show 261 1049 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 884 1049 moveto (\)) show end pse gR gS 17 23 560 797 rC 383 554 :M f56 sf -.032( is also the statement, that we)A 70 575 :M .978 .098(make here in the simplest possible context of a particle moving on the plane )J 459 563 17 13 rC 459 576 :M f133 sf ( )S 476 576 :M psb currentpoint pse 459 563 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 544 div 416 3 -1 roll exch div scale currentpoint translate 64 -630 translate -32 1014 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Math f1 (R) show 314 843 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 476 575 :M f56 sf 1.843 .184(, that the)J 70 592 :M -.137(random variable, call it )A 181 579 55 20 rC 236 599 :M psb currentpoint pse 181 579 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1760 div 640 3 -1 roll exch div scale currentpoint translate 64 -1199 translate -4 1615 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 1711 moveto 224 /Times-Roman f1 (3) show 522 1615 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 939 1615 moveto 384 /Times-Italic f1 (t) show 1073 1711 moveto 224 /Times-Roman f1 (0) show 1219 1615 moveto 384 /Times-Roman f1 (,) show 1334 1615 moveto 384 /Symbol f1 (\327) show 796 1645 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1419 1645 moveto (\)) show 388 1662 moveto 384 1000 1580 /Symbol f3 (\() show 1546 1662 moveto (\)) show end pse gR gS 17 23 560 797 rC 236 592 :M f56 sf -.109(, defined by having characteristic function)A 96 600 400 35 rC 496 635 :M psb currentpoint pse 96 600 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12800 div 1120 3 -1 roll exch div scale currentpoint translate 64 -1114 translate -4 1754 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) show 244 1905 moveto 224 /Times-Italic f1 (L) show 402 1961 moveto 160 /Times-Roman f1 (3) show 614 1905 moveto 224 /Symbol f2 (j) show 885 1905 moveto 224 /Times-Italic f1 (t) show 976 1961 moveto 160 /Times-Roman f1 (0) show 1102 1905 moveto 224 /Times-Roman f1 (,) show 1164 1905 moveto 224 /Symbol f1 (\327) show 788 1923 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 224 1000 1383 /Symbol f3 (\() show 1227 1923 moveto (\)) show 523 1933 moveto 224 1000 1571 /Symbol f3 (\() show 1315 1933 moveto (\)) show 1565 1754 moveto 384 /Symbol f2 (t) show 1421 1764 moveto 384 1000 1171 /Symbol f3 (\() show 1784 1764 moveto (\)) show 2014 1754 moveto 384 /Symbol f1 (=) show 2710 1754 moveto 384 /Times-Italic f1 (r) show 2908 1754 moveto 384 /Symbol f1 (\327) show 3059 1754 moveto 384 /Times-Italic f1 (dr) show 3711 1754 moveto (d) show 3903 1754 moveto 384 /Symbol f2 (J) show 3518 2173 moveto 224 /Times-Roman f1 (0) show 3405 1305 moveto (2) show 3528 1305 moveto 224 /Symbol f1 (\327) show 3595 1305 moveto 224 /Symbol f2 (p) show 3489 1888 moveto 576 /Symbol f1 (\362) show 4189 1754 moveto 384 /Times-Roman f1 ( ) show 4400 1754 moveto 384 /Symbol f2 (j) show 4817 1754 moveto 384 /Times-Italic f1 (t) show 4951 1850 moveto 224 /Times-Roman f1 (0) show 5097 1754 moveto 384 /Times-Roman f1 (,) show 5232 1754 moveto 384 /Times-Italic f1 (x) show 5423 1850 moveto 224 /Times-Roman f1 (1) show 5701 1754 moveto 384 /Times-Italic f1 (r) show 5837 1754 moveto 384 /Times-Roman f1 (,) show 5972 1754 moveto 384 /Symbol f2 (J) show 5555 1766 moveto 384 1000 1190 /Symbol f3 (\() show 6279 1766 moveto (\)) show 6393 1754 moveto 384 /Times-Roman f1 (,) show 6528 1754 moveto 384 /Times-Italic f1 (x) show 6737 1850 moveto 224 /Times-Roman f1 (2) show 7033 1754 moveto 384 /Times-Italic f1 (r) show 7169 1754 moveto 384 /Times-Roman f1 (,) show 7304 1754 moveto 384 /Symbol f2 (J) show 6887 1766 moveto 384 1000 1190 /Symbol f3 (\() show 7611 1766 moveto (\)) show 4674 1784 moveto 384 1000 1388 /Symbol f3 (\() show 7738 1784 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4413 1354 moveto 3431 0 rlineto stroke 2442 2173 moveto 224 /Times-Roman f1 (0) show 2346 1305 moveto 224 /Symbol f1 (+) show 2493 1305 moveto (\245) show 2413 1888 moveto 576 ns (\362) show 7914 1754 moveto 384 ns (\327) show 8061 1754 moveto 384 /Symbol f2 (j) show 8478 1754 moveto 384 /Times-Italic f1 (t) show 8612 1850 moveto 224 /Times-Roman f1 (0) show 8758 1754 moveto 384 /Times-Roman f1 (,) show 8893 1754 moveto 384 /Times-Italic f1 (x) show 9084 1850 moveto 224 /Times-Roman f1 (1) show 9362 1754 moveto 384 /Times-Italic f1 (r) show 9498 1754 moveto 384 /Times-Roman f1 (,) show 9633 1754 moveto 384 /Symbol f2 (J) show 10012 1754 moveto 384 /Symbol f1 (+) show 10311 1754 moveto 384 /Symbol f2 (t) show 9216 1766 moveto 384 1000 1190 /Symbol f3 (\() show 10530 1766 moveto (\)) show 10644 1754 moveto 384 /Times-Roman f1 (,) show 10779 1754 moveto 384 /Times-Italic f1 (x) show 10988 1850 moveto 224 /Times-Roman f1 (2) show 11284 1754 moveto 384 /Times-Italic f1 (r) show 11420 1754 moveto 384 /Times-Roman f1 (,) show 11555 1754 moveto 384 /Symbol f2 (J) show 11934 1754 moveto 384 /Symbol f1 (+) show 12233 1754 moveto 384 /Symbol f2 (t) show 11138 1766 moveto 384 1000 1190 /Symbol f3 (\() show 12452 1766 moveto (\)) show 8335 1784 moveto 384 1000 1388 /Symbol f3 (\() show 12579 1784 moveto (\)) show end pse gR gS 17 23 560 797 rC 70 648 :M f56 sf .374 .037(has some relevance in the description of the behaviour of the system described at time )J 495 638 11 15 rC 506 653 :M psb currentpoint pse 495 638 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 352 div 480 3 -1 roll exch div scale currentpoint translate -344 -8 translate 404 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 538 424 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 506 648 :M f56 sf 1.667 .167( by)J 70 654 34 18 rC 104 672 :M psb currentpoint pse 70 654 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 576 3 -1 roll exch div scale currentpoint translate 64 -635 translate -13 1019 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 1019 moveto 384 /Times-Italic f1 (t) show 538 1115 moveto 224 /Times-Roman f1 (0) show 684 1019 moveto 384 /Times-Roman f1 (,) show 799 1019 moveto 384 /Symbol f1 (\327) show 261 1049 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 884 1049 moveto (\)) show end pse gR gS 17 23 560 797 rC 104 666 :M f56 sf .379 .038( and that, for "some" reason, it deserves the name of "orbital angular momentum with)J 70 687 :M -.069(respect to the origin".)A 70 721 :M .74 .074(The question we raise in this paper is the following: having fixed )J 399 711 11 15 rC 410 726 :M psb currentpoint pse 399 711 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 352 div 480 3 -1 roll exch div scale currentpoint translate -344 -8 translate 404 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 538 424 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 410 721 :M f56 sf 1.292 .129(, having fixed )J 485 709 34 18 rC 519 727 :M psb currentpoint pse 485 709 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 576 3 -1 roll exch div scale currentpoint translate 64 -635 translate -13 1019 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 404 1019 moveto 384 /Times-Italic f1 (t) show 538 1115 moveto 224 /Times-Roman f1 (0) show 684 1019 moveto 384 /Times-Roman f1 (,) show 799 1019 moveto 384 /Symbol f1 (\327) show 261 1049 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 884 1049 moveto (\)) show end pse gR gS 17 23 560 797 rC 519 721 :M f56 sf (,)S 70 740 :M -.019(having determined, according to the usual procedure, the current velocity field )A 448 728 36 18 rC 484 746 :M psb currentpoint pse 448 728 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1152 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -3 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 331 328 moveto (t) show 465 424 moveto 224 /Times-Roman f1 (0) show 611 328 moveto 384 /Times-Roman f1 (,) show 746 328 moveto 384 /Times-Italic f1 (x) show 188 358 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 950 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 484 740 :M f56 sf .375 .038( and the)J 70 759 :M -.012(probability density field )A 188 747 37 18 rC 225 765 :M psb currentpoint pse 188 747 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1184 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -19 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (r) show 361 328 moveto 384 /Times-Italic f1 (t) show 495 424 moveto 224 /Times-Roman f1 (0) show 641 328 moveto 384 /Times-Roman f1 (,) show 776 328 moveto 384 /Times-Italic f1 (x) show 218 358 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 980 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 225 759 :M f56 sf .71 .071(, is there a diffusion process )J 369 747 61 18 rC 430 765 :M psb currentpoint pse 369 747 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1952 div 576 3 -1 roll exch div scale currentpoint translate 64 -1199 translate 153 1583 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 499 1583 moveto (t) show 356 1593 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 627 1593 moveto (\)) show 741 1583 moveto 384 /Times-Roman f1 (,) show 878 1583 moveto 384 /Times-Italic f1 (t) show 1086 1583 moveto 384 /Symbol f1 (\263) show 1395 1583 moveto 384 /Times-Italic f1 (t) show 1529 1679 moveto 224 /Times-Roman f1 (0) show -22 1616 moveto 384 1000 1401 /Symbol f3 ({) show 1669 1616 moveto (}) show end pse gR gS 17 23 560 797 rC 430 759 :M f56 sf .856 .086( evolving from the)J endp showpage %%Page: 7 7 %%BeginPageSetup initializepage (defalco; pag: 7 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (7)S 70 98 :M -.171(initial condition determined by )A 216 86 36 18 rC 252 104 :M psb currentpoint pse 216 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1152 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -3 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 331 328 moveto (t) show 465 424 moveto 224 /Times-Roman f1 (0) show 611 328 moveto 384 /Times-Roman f1 (,) show 746 328 moveto 384 /Times-Italic f1 (x) show 188 358 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 950 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 252 98 :M f56 sf -.081( and )A 275 86 37 18 rC 312 104 :M psb currentpoint pse 275 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1184 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -19 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (r) show 361 328 moveto 384 /Times-Italic f1 (t) show 495 424 moveto 224 /Times-Roman f1 (0) show 641 328 moveto 384 /Times-Roman f1 (,) show 776 328 moveto 384 /Times-Italic f1 (x) show 218 358 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 980 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 312 98 :M f56 sf -.044(, a suitable functional of which has the same)A 70 118 :M 2.186 .219(law as )J 109 105 55 20 rC 164 125 :M psb currentpoint pse 109 105 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1760 div 640 3 -1 roll exch div scale currentpoint translate 64 -1199 translate -4 1615 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 1711 moveto 224 /Times-Roman f1 (3) show 522 1615 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 939 1615 moveto 384 /Times-Italic f1 (t) show 1073 1711 moveto 224 /Times-Roman f1 (0) show 1219 1615 moveto 384 /Times-Roman f1 (,) show 1334 1615 moveto 384 /Symbol f1 (\327) show 796 1645 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1419 1645 moveto (\)) show 388 1662 moveto 384 1000 1580 /Symbol f3 (\() show 1546 1662 moveto (\)) show end pse gR gS 17 23 560 797 rC 164 118 :M f56 sf 2.606 .261( ? Having found such a process )J 343 106 61 18 rC 404 124 :M psb currentpoint pse 343 106 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1952 div 576 3 -1 roll exch div scale currentpoint translate 64 -1199 translate 153 1583 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 499 1583 moveto (t) show 356 1593 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 627 1593 moveto (\)) show 741 1583 moveto 384 /Times-Roman f1 (,) show 878 1583 moveto 384 /Times-Italic f1 (t) show 1086 1583 moveto 384 /Symbol f1 (\263) show 1395 1583 moveto 384 /Times-Italic f1 (t) show 1529 1679 moveto 224 /Times-Roman f1 (0) show -22 1616 moveto 384 1000 1401 /Symbol f3 ({) show 1669 1616 moveto (}) show end pse gR gS 17 23 560 797 rC 404 118 :M f56 sf 2.247 .225( and such a functional)J 70 126 70 18 rC 140 144 :M psb currentpoint pse 70 126 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2240 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -4 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 375 328 moveto (q) show 721 328 moveto (t) show 578 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 849 338 moveto (\)) show 987 328 moveto 384 /Times-Roman f1 ( ) show 1091 328 moveto 384 /Times-Roman f1 (,) show 1228 328 moveto 384 /Times-Italic f1 (t) show 1436 328 moveto 384 /Symbol f1 (\263) show 1745 328 moveto 384 /Times-Italic f1 (t) show 1879 424 moveto 224 /Times-Roman f1 (0) show 237 358 moveto 384 1000 1388 /Symbol f3 (\() show 2038 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 140 138 :M f56 sf 1.74 .174( does the "force" )J 233 126 59 17 rC 292 143 :M psb currentpoint pse 233 126 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1888 div 544 3 -1 roll exch div scale currentpoint translate 64 -1093 translate -4 1477 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show 359 1477 moveto 384 /Symbol f1 (\327) show 505 1477 moveto 384 /Times-Italic f1 (a) show 829 1477 moveto (t) show 944 1477 moveto 384 /Times-Roman f1 (,) show 1076 1477 moveto 384 /Times-Italic f1 (q) show 1422 1477 moveto (t) show 1279 1487 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1550 1487 moveto (\)) show 686 1505 moveto 384 1000 1364 /Symbol f3 (\() show 1677 1505 moveto (\)) show end pse gR gS 17 23 560 797 rC 292 138 :M f56 sf 1.585 .158( under which this process evolves justify the)J 70 158 :M 1.375 .138(attribution to )J 140 146 70 18 rC 210 164 :M psb currentpoint pse 140 146 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2240 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -4 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 375 328 moveto (q) show 721 328 moveto (t) show 578 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 849 338 moveto (\)) show 987 328 moveto 384 /Times-Roman f1 ( ) show 1091 328 moveto 384 /Times-Roman f1 (,) show 1228 328 moveto 384 /Times-Italic f1 (t) show 1436 328 moveto 384 /Symbol f1 (\263) show 1745 328 moveto 384 /Times-Italic f1 (t) show 1879 424 moveto 224 /Times-Roman f1 (0) show 237 358 moveto 384 1000 1388 /Symbol f3 (\() show 2038 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 210 158 :M f56 sf 2.594 .259( and, therefore to )J 310 145 55 20 rC 365 165 :M psb currentpoint pse 310 145 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1760 div 640 3 -1 roll exch div scale currentpoint translate 64 -1199 translate -4 1615 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 1711 moveto 224 /Times-Roman f1 (3) show 522 1615 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 939 1615 moveto 384 /Times-Italic f1 (t) show 1073 1711 moveto 224 /Times-Roman f1 (0) show 1219 1615 moveto 384 /Times-Roman f1 (,) show 1334 1615 moveto 384 /Symbol f1 (\327) show 796 1645 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1419 1645 moveto (\)) show 388 1662 moveto 384 1000 1580 /Symbol f3 (\() show 1546 1662 moveto (\)) show end pse gR gS 17 23 560 797 rC 365 158 :M f56 sf 2.173 .217(, of the name "orbital angular)J 70 180 :M 2.273 .227(momentum with respect to the origin" ? By this we mean: under )J f78 sf 12 u78 :p 46.051 :m .601(classical)A f56 sf 3.386 .339( evolution)J 70 184 68 18 rC 138 202 :M psb currentpoint pse 70 184 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2176 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate 156 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 494 328 moveto (t) show 351 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 622 338 moveto (\)) show 760 328 moveto 384 /Times-Roman f1 ( ) show 864 328 moveto 384 /Times-Roman f1 (,) show 1005 328 moveto 384 /Times-Roman f1 ( ) show 1101 328 moveto 384 /Times-Italic f1 (t) show 1309 328 moveto 384 /Symbol f1 (\263) show 1618 328 moveto 384 /Times-Italic f1 (t) show 1752 424 moveto 224 /Times-Roman f1 (0) show -22 361 moveto 384 1000 1401 /Symbol f3 ({) show 1892 361 moveto (}) show end pse gR gS 17 23 560 797 rC 138 196 :M f56 sf .425 .043( under the force )J 219 184 59 17 rC 278 201 :M psb currentpoint pse 219 184 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1888 div 544 3 -1 roll exch div scale currentpoint translate 64 -1093 translate -4 1477 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show 359 1477 moveto 384 /Symbol f1 (\327) show 505 1477 moveto 384 /Times-Italic f1 (a) show 829 1477 moveto (t) show 944 1477 moveto 384 /Times-Roman f1 (,) show 1079 1477 moveto 384 /Times-Italic f1 (x) show 1417 1477 moveto (t) show 1274 1487 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1545 1487 moveto (\)) show 686 1505 moveto 384 1000 1364 /Symbol f3 (\() show 1672 1505 moveto (\)) show end pse gR gS 17 23 560 797 rC 278 196 :M f56 sf 1.926 .193(, does )J 314 184 70 18 rC 384 202 :M psb currentpoint pse 314 184 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2240 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -4 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 378 328 moveto (x) show 716 328 moveto (t) show 573 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 844 338 moveto (\)) show 982 328 moveto 384 /Times-Roman f1 ( ) show 1086 328 moveto 384 /Times-Roman f1 (,) show 1223 328 moveto 384 /Times-Italic f1 (t) show 1431 328 moveto 384 /Symbol f1 (\263) show 1740 328 moveto 384 /Times-Italic f1 (t) show 1874 424 moveto 224 /Times-Roman f1 (0) show 237 358 moveto 384 1000 1388 /Symbol f3 (\() show 2033 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 384 196 :M f56 sf .873 .087( assume the value of orbital)J 70 219 :M -.169(angular momentum competing to the initial condition )A 321 203 58 22 rC 379 225 :M psb currentpoint pse 321 203 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1856 div 704 3 -1 roll exch div scale currentpoint translate 64 -1110 translate -6 1622 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 332 1622 moveto (t) show 466 1718 moveto 224 /Times-Roman f1 (0) show 189 1652 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 625 1652 moveto (\)) show 763 1622 moveto 384 /Times-Roman f1 ( ) show 867 1622 moveto 384 /Times-Roman f1 (,) show 1002 1622 moveto 384 /Times-Italic f1 (x) show 1045 1305 moveto 224 /Symbol f1 (\267) show 1340 1622 moveto 384 /Times-Italic f1 (t) show 1474 1718 moveto 224 /Times-Roman f1 (0) show 1197 1652 moveto 384 1000 1388 /Symbol f3 (\() show 1633 1652 moveto (\)) show end pse gR gS 17 23 560 797 rC 379 219 :M f56 sf -.326( ?)A 72 258 :M .411 .041(Figure 1 shows an example of the fact that the above questions admit a positive answer; in)J 72 276 :M -.044(particular it shows that there do exist integer-valued functionals of Nelson's processes. It refers)A 72 294 :M -.122(to a numerical simulation of the stochastic process)A 238 298 119 18 rC 357 316 :M psb currentpoint pse 238 298 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3808 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -9 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 328 moveto (t) show 194 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 338 moveto (\)) show 679 328 moveto 384 /Symbol f1 (=) show 1115 328 moveto 384 /Times-Italic f1 (q) show 1314 424 moveto 224 /Times-Roman f1 (1) show 1589 328 moveto 384 /Times-Italic f1 (t) show 1446 338 moveto 384 1000 1171 /Symbol f3 (\() show 1717 338 moveto (\)) show 1831 328 moveto 384 /Times-Roman f1 (,) show 1963 328 moveto 384 /Times-Italic f1 (q) show 2180 424 moveto 224 /Times-Roman f1 (2) show 2473 328 moveto 384 /Times-Italic f1 (t) show 2330 338 moveto 384 1000 1171 /Symbol f3 (\() show 2601 338 moveto (\)) show 2715 328 moveto 384 /Times-Roman f1 (,) show 2847 328 moveto 384 /Times-Italic f1 (q) show 3061 424 moveto 224 /Times-Roman f1 (3) show 3344 328 moveto 384 /Times-Italic f1 (t) show 3201 338 moveto 384 1000 1171 /Symbol f3 (\() show 3472 338 moveto (\)) show 977 358 moveto 384 1000 1388 /Symbol f3 (\() show 3599 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 331 :M f56 sf -.02(corresponding, in the sense of Nelson [1, 2, 3], to a Schr\232dinger wave function evolving in a)A 72 347 :M -.161(magnetic field having a gradient in the )A 254 337 13 15 rC 267 352 :M psb currentpoint pse 254 337 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont 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3 1 roll sub 1216 div 992 3 -1 roll exch div scale currentpoint translate 64 45 translate 158 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 394 388 moveto 224 /Times-Roman f1 (3) show 677 292 moveto 384 /Times-Italic f1 (t) show 534 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 833 302 moveto (\)) show 24 824 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 295 824 moveto 384 /Symbol f1 (\327) show 451 824 moveto 384 /Times-Italic f1 (t) show 612 653 moveto 224 /Times-Roman f1 (2) show 893 824 moveto 384 ns (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 886 501 vb 16 th 0 432 moveto 1107 0 rlineto stroke end pse gR gS 17 23 560 797 rC 272 577 :M f193 sf -.078( as a function of )A 351 569 7 10 rC 358 579 :M psb currentpoint pse 351 569 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -606 -36 translate 666 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 72 605 :M f56 sf .786 .079(We shall give a detailed description of the model - an idealised version of a Stern-Gerlach)J 72 623 :M .522 .052(apparatus [4] - in Section 2. Here we wish to point out that Figure 1 shows that the average)J 72 639 :M -.246(acceleration )A 130 627 51 17 rC 181 644 :M psb currentpoint pse 130 627 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1632 div 544 3 -1 roll exch div scale currentpoint translate 64 -630 translate -6 1014 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (~) show 297 1014 moveto 384 /Times-Italic f1 (q) show 511 1110 moveto 224 /Times-Roman f1 (3) show 794 1014 moveto 384 /Times-Italic f1 (t) show 651 1024 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 922 1024 moveto (\)) show 1236 1014 moveto 384 /Times-Italic f1 (t) show 1369 843 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -136 422 1194 691 vb end pse gR gS 17 23 560 797 rC 181 639 :M f56 sf -.03( in the direction of the field gradient tends to a discrete random variable)A 72 659 :M .839 .084(as )J 86 651 7 10 rC 93 661 :M psb currentpoint pse 86 651 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -2497 -72 translate 2557 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 93 659 :M f56 sf .099 .01( increases; this fact closely parallels, in the case of orbital angular momentum considered)J 72 677 :M 1.374 .137(here, the results obtained by Faris [5] in the case of spin angular momentum. As only a)J 72 693 :M -.011(configurational observable )A 203 682 26 16 rC 229 698 :M psb currentpoint pse 203 682 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate -2005 24 translate 2060 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 2274 424 moveto 224 /Times-Roman f1 (3) show 2557 328 moveto 384 /Times-Italic f1 (t) show 2414 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 2685 338 moveto (\)) show end pse gR gS 17 23 560 797 rC 230 693 :M f56 sf 1.023 .102( is involved in this phenomenon, and as the predictions of)J 72 713 :M .649 .065(quantum mechanics and stochastic mechanics coincide on such an observable, it will be, in)J 72 731 :M .268 .027(Section 2, a simple exercise in quantum mechanics to prove, in our context, the generality of)J 72 749 :M -.066(the limiting behaviour shown in Figure 1.)A endp showpage %%Page: 8 8 %%BeginPageSetup initializepage (defalco; pag: 8 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (8)S 72 98 :M .194 .019(The configuration space for the process we consider is )J 339 86 16 13 rC 339 99 :M f133 sf ( )S 355 99 :M psb currentpoint pse 339 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 416 3 -1 roll exch div scale currentpoint translate 64 61 translate -32 323 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Math f1 (R) show 311 152 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 355 98 :M f56 sf .851 .085(, as opposed to )J 434 86 58 16 rC 434 102 :M f133 sf ( )S 492 102 :M psb currentpoint pse 434 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1856 div 512 3 -1 roll exch div scale currentpoint translate 64 61 translate -32 323 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Math f1 (R) show 311 152 moveto 224 /Times-Roman f1 (3) show 529 323 moveto 384 /Symbol f1 (\264) show 813 323 moveto 384 /Times-Italic f1 (SU) show 1444 323 moveto 384 /Times-Roman f1 (2) show 1308 333 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1647 333 moveto (\)) show end pse gR gS 17 23 560 797 rC 492 98 :M f56 sf .387 .039( in the)J 72 117 :M .112 .011(case of Faris [5]. The simplicity we gain in this more elementary context allows us to explore)J 72 133 :M -.024(the question whether there are, beyond )A 260 121 41 17 rC 301 138 :M psb currentpoint pse 260 121 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1312 div 544 3 -1 roll exch div scale currentpoint translate 64 61 translate -9 323 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 205 419 moveto 224 /Times-Roman f1 (3) show 488 323 moveto 384 /Times-Italic f1 (t) show 345 333 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 616 333 moveto (\)) show 930 323 moveto 384 /Times-Italic f1 (t) show 1063 152 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -136 422 888 0 vb end pse gR gS 17 23 560 797 rC 302 133 :M f56 sf .259 .026(, other functionals of the processes exhibiting)J 72 153 :M -.079(the same asymptotic behaviour. Figure 2 shows that this is the case: it follows the motion in the)A 72 160 27 14 rC 99 174 :M psb currentpoint pse 72 160 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 357 moveto 224 /Times-Roman f1 (1) show 313 261 moveto 384 /Times-Roman f1 (,) show 448 261 moveto 384 /Times-Italic f1 (x) show 657 357 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 99 170 :M f56 sf -.138( plane in terms of the time dependence of a stochastic analogue)A 195 187 :M ( )S 198 175 202 17 rC 400 192 :M psb currentpoint pse 198 175 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6464 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 324 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) show 243 420 moveto 224 /Times-Roman f1 (3) show 526 324 moveto 384 /Times-Italic f1 (t) show 383 334 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 654 334 moveto (\)) show 868 324 moveto 384 /Symbol f1 (=) show 1177 324 moveto 384 /Times-Italic f1 (q) show 1376 420 moveto 224 /Times-Roman f1 (1) show 1651 324 moveto 384 /Times-Italic f1 (t) show 1508 334 moveto 384 1000 1171 /Symbol f3 (\() show 1779 334 moveto (\)) show 1955 324 moveto 384 /Symbol f1 (\327) show 2115 324 moveto 384 /Times-Italic f1 (p) show 2321 420 moveto 224 /Times-Roman f1 (2) show 2614 324 moveto 384 /Times-Italic f1 (t) show 2729 324 moveto 384 /Times-Roman f1 (,) show 2861 324 moveto 384 /Times-Italic f1 (q) show 3207 324 moveto (t) show 3064 334 moveto 384 1000 1171 /Symbol f3 (\() show 3335 334 moveto (\)) show 2471 352 moveto 384 1000 1364 /Symbol f3 (\() show 3462 352 moveto (\)) show 3660 324 moveto 384 /Symbol f1 (-) show 3952 324 moveto 384 /Times-Italic f1 (q) show 4169 420 moveto 224 /Times-Roman f1 (2) show 4462 324 moveto 384 /Times-Italic f1 (t) show 4319 334 moveto 384 1000 1171 /Symbol f3 (\() show 4590 334 moveto (\)) show 4766 324 moveto 384 /Symbol f1 (\327) show 4926 324 moveto 384 /Times-Italic f1 (p) show 5114 420 moveto 224 /Times-Roman f1 (1) show 5389 324 moveto 384 /Times-Italic f1 (t) show 5504 324 moveto 384 /Times-Roman f1 (,) show 5636 324 moveto 384 /Times-Italic f1 (q) show 5982 324 moveto (t) show 5839 334 moveto 384 1000 1171 /Symbol f3 (\() show 6110 334 moveto (\)) show 5246 352 moveto 384 1000 1364 /Symbol f3 (\() show 6237 352 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 205 :M f56 sf -.109(of the )A 102 195 13 15 rC 115 210 :M psb currentpoint pse 102 195 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 223 357 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 115 205 :M f56 sf -.089( component of classical canonical angular momentum and shows that also )A 467 194 27 16 rC 494 210 :M psb currentpoint pse 467 194 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 512 3 -1 roll exch div scale currentpoint translate 64 -1251 translate -9 1603 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) show 243 1699 moveto 224 /Times-Roman f1 (3) show 526 1603 moveto 384 /Times-Italic f1 (t) show 383 1613 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 654 1613 moveto (\)) show end pse gR gS 17 23 560 797 rC 494 205 :M f56 sf .449 .045( tends)J 72 225 :M .032 .003(to a discrete random variable. Section 3 will be devoted to a construction, in a simple context,)J 72 241 :M .538 .054(of the process )J 144 230 21 15 rC 165 245 :M psb currentpoint pse 144 230 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate 64 24 translate -9 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 328 moveto (t) show 194 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 338 moveto (\)) show end pse gR gS 17 23 560 797 rC 165 241 :M f56 sf .839 .084( and to the definition of the random variables )J 396 229 48 18 rC 444 247 :M psb currentpoint pse 396 229 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1536 div 576 3 -1 roll exch div scale currentpoint translate 64 -623 translate 0 1007 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 224 1103 moveto 224 ns (j) show 458 1007 moveto 384 ns (t) show 573 1007 moveto 384 /Times-Roman f1 (,) show 705 1007 moveto 384 /Times-Italic f1 (q) show 1051 1007 moveto (t) show 908 1017 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1179 1017 moveto (\)) show 315 1035 moveto 384 1000 1364 /Symbol f3 (\() show 1306 1035 moveto (\)) show end pse gR gS 17 23 560 797 rC 444 241 :M f56 sf 1.156 .116( in terms of the)J 72 261 :M -.138(current velocity field associated with )A 247 250 21 15 rC 268 265 :M psb currentpoint pse 247 250 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate 64 24 translate -9 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 328 moveto (t) show 194 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 338 moveto (\)) show end pse gR gS 17 23 560 797 rC 268 261 :M f56 sf -.141( and of the vector potential )A 396 250 26 15 rC 422 265 :M psb currentpoint pse 396 250 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -5 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (A) show 419 292 moveto (x) show 278 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 623 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 422 261 :M f56 sf (.)S 152 284 282 174 rC 282 174 8 846 [282 0 0 174 152 284] F T 1 F :f :l FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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neg 3 1 roll sub 5280 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 324 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 190 420 moveto 224 /Times-Roman f1 (1) show 465 324 moveto 384 /Times-Italic f1 (t) show 322 334 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 593 334 moveto (\)) show 769 324 moveto 384 /Symbol f1 (\327) show 929 324 moveto 384 /Times-Italic f1 (p) show 1135 420 moveto 224 /Times-Roman f1 (2) show 1428 324 moveto 384 /Times-Italic f1 (t) show 1543 324 moveto 384 /Times-Roman f1 (,) show 1675 324 moveto 384 /Times-Italic f1 (q) show 2021 324 moveto (t) show 1878 334 moveto 384 1000 1171 /Symbol f3 (\() show 2149 334 moveto (\)) show 1285 352 moveto 384 1000 1364 /Symbol f3 (\() show 2276 352 moveto (\)) show 2474 324 moveto 384 /Symbol f1 (-) show 2766 324 moveto 384 /Times-Italic f1 (q) show 2983 420 moveto 224 /Times-Roman f1 (2) show 3276 324 moveto 384 /Times-Italic f1 (t) show 3133 334 moveto 384 1000 1171 /Symbol f3 (\() show 3404 334 moveto (\)) show 3580 324 moveto 384 /Symbol f1 (\327) show 3740 324 moveto 384 /Times-Italic f1 (p) show 3928 420 moveto 224 /Times-Roman f1 (1) show 4203 324 moveto 384 /Times-Italic f1 (t) show 4318 324 moveto 384 /Times-Roman f1 (,) show 4450 324 moveto 384 /Times-Italic f1 (q) show 4796 324 moveto (t) show 4653 334 moveto 384 1000 1171 /Symbol f3 (\() show 4924 334 moveto (\)) show 4060 352 moveto 384 1000 1364 /Symbol f3 (\() show 5051 352 moveto (\)) show end pse gR gS 17 23 560 797 rC 336 484 :M f193 sf -.078( as a function of )A 415 476 7 10 rC 422 486 :M psb currentpoint pse 415 476 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -606 -36 translate 666 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 72 516 :M f56 sf -.055(In Section 4 we shall comment on the line of reasoning leading to the previous observations, in)A 72 534 :M 2.356 .236(the framework of the following statement by R. Feynman and A. Hibbs: "Indeed all)J 72 552 :M -.017(measurements of quantum mechanical systems could be made to reduce eventually to position)A 72 570 :M .985 .099(and time measurements \(e.g. the position of a needle on a meter or the time of flight of a)J 72 588 :M .081 .008(particle\). Because of this possibility a theory formulated in terms of position measurements is)J 72 606 :M .664 .066(complete enough in principle to describe all phenomena" [6, p. 96]. We shall argue that, in)J 72 624 :M -.029(modelling the measurement of a component of orbital angular momentum, the obvious pointer)A 72 640 :M -.205(variable )A 112 628 41 17 rC 153 645 :M psb currentpoint pse 112 628 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1312 div 544 3 -1 roll exch div scale currentpoint translate -242 61 translate 297 323 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 511 419 moveto 224 /Times-Roman f1 (3) show 794 323 moveto 384 /Times-Italic f1 (t) show 651 333 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 922 333 moveto (\)) show 1236 323 moveto 384 /Times-Italic f1 (t) show 1369 152 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -136 422 1194 0 vb end pse gR gS 17 23 560 797 rC 153 640 :M f56 sf -.054( is not the only configurational observable that conforms to the above point of)A 72 658 :M .782 .078(view; we shall give, indeed, evidence of the fact that, though )J 380 647 27 16 rC 407 663 :M psb currentpoint pse 380 647 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 512 3 -1 roll exch div scale currentpoint translate 64 -1251 translate -9 1603 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) show 243 1699 moveto 224 /Times-Roman f1 (3) show 526 1603 moveto 384 /Times-Italic f1 (t) show 383 1613 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 654 1613 moveto (\)) show end pse gR gS 17 23 560 797 rC 407 658 :M f56 sf .765 .076( makes reference to the)J 72 678 :M -.005(current velocity field of the process, its limit as )A 301 670 40 10 rC 341 680 :M psb currentpoint pse 301 670 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1280 div 320 3 -1 roll exch div scale currentpoint translate 64 -5 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 197 261 moveto 384 /Symbol f1 (\256) show 674 261 moveto (+) show 903 261 moveto (\245) show end pse gR gS 17 23 560 797 rC 341 678 :M f56 sf .159 .016( coincides with the limit of the purely)J 72 696 :M -.153(configurational random variable)A 100 718 395 28 rC 495 746 :M psb currentpoint pse 100 718 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 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/Times-Italic f1 (dq) show 3108 596 moveto 224 /Times-Roman f1 (2) show 3386 500 moveto 384 /Times-Italic f1 (s) show 3258 510 moveto 384 1000 1171 /Symbol f3 (\() show 3540 510 moveto (\)) show 3738 500 moveto 384 /Symbol f1 (-) show 4030 500 moveto 384 /Times-Italic f1 (q) show 4247 596 moveto 224 /Times-Roman f1 (2) show 4525 500 moveto 384 /Times-Italic f1 (s) show 4397 510 moveto 384 1000 1171 /Symbol f3 (\() show 4679 510 moveto (\)) show 4855 500 moveto 384 /Symbol f1 (\327) show 5006 500 moveto 384 /Times-Italic f1 (dq) show 5397 596 moveto 224 /Times-Roman f1 (1) show 5657 500 moveto 384 /Times-Italic f1 (s) show 5529 510 moveto 384 1000 1171 /Symbol f3 (\() show 5811 510 moveto (\)) show 1598 714 moveto 224 /Times-Roman f1 (0) show 1658 202 moveto 224 /Times-Italic f1 (t) show 1461 634 moveto 576 /Symbol f1 (\362) show 6010 500 moveto 384 ns (+) show 6302 261 moveto 384 /Times-Roman f1 (1) show 6343 793 moveto 384 /Times-Italic f1 (t) show 6312 401 moveto 173 0 rlineto stroke 6973 500 moveto (q) show 7172 596 moveto 224 /Times-Roman f1 (1) show 7432 500 moveto 384 /Times-Italic f1 (s) show 7304 510 moveto 384 1000 1171 /Symbol f3 (\() show 7586 510 moveto (\)) show 7762 500 moveto 384 /Symbol f1 (\327) show 7917 500 moveto 384 /Times-Italic f1 (A) show 8214 596 moveto 224 /Times-Roman f1 (2) show 8502 500 moveto 384 /Times-Italic f1 (q) show 8833 500 moveto (s) show 8705 510 moveto 384 1000 1171 /Symbol f3 (\() show 8987 510 moveto (\)) show 8364 528 moveto 384 1000 1364 /Symbol f3 (\() show 9114 528 moveto (\)) show 9312 500 moveto 384 /Symbol f1 (-) show 9604 500 moveto 384 /Times-Italic f1 (q) show 9821 596 moveto 224 /Times-Roman f1 (2) show 10099 500 moveto 384 /Times-Italic f1 (s) show 9971 510 moveto 384 1000 1171 /Symbol f3 (\() show 10253 510 moveto (\)) show 10429 500 moveto 384 /Symbol f1 (\327) show 10584 500 moveto 384 /Times-Italic f1 (A) show 10863 596 moveto 224 /Times-Roman f1 (1) show 11133 500 moveto 384 /Times-Italic f1 (q) show 11464 500 moveto (s) show 11336 510 moveto 384 1000 1171 /Symbol f3 (\() show 11618 510 moveto (\)) show 10995 528 moveto 384 1000 1364 /Symbol f3 (\() show 11745 528 moveto (\)) show 6835 545 moveto 384 1000 1557 /Symbol f3 (\() show 11872 545 moveto (\)) show 12048 500 moveto 384 /Symbol f1 (\327) show 12199 500 moveto 384 /Times-Italic f1 (ds) show 6685 714 moveto 224 /Times-Roman f1 (0) show 6745 202 moveto 224 /Times-Italic f1 (t) show 6548 634 moveto 576 /Symbol f1 (\362) show end pse endp showpage %%Page: 9 9 %%BeginPageSetup initializepage (defalco; pag: 9 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 294 802 :M f56 sf (9)S 72 94 :M .292 .029(This last observation supports the conjecture, advanced in [7] in an attempt of understanding)J 72 112 :M -.018(Bohr quantization in the stochastic context, that in Nelson's stochastic mechanics one can give)A 72 130 :M -.101(a Keplerian definition of angular momentum in terms of "stochastic area per unit time", with, of)A 72 148 :M -.097(course, the purely classical correction given by the field dependent addendum.)A 72 187 :M f67 sf .562 .056(2. The model)J 72 205 :M f56 sf -.132(The quantum mechanical predictions on the result of a measurement of the observable)A 236 227 122 36 rC 358 263 :M psb currentpoint pse 236 227 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3904 div 1152 3 -1 roll exch div scale currentpoint translate 64 53 translate -4 619 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 715 moveto 224 /Times-Roman f1 (3) show 484 619 moveto 384 /Symbol f1 (=) show 792 380 moveto 384 /Times-Roman f1 (1) show 835 912 moveto 384 /Times-Italic f1 (i) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 802 520 moveto 173 0 rlineto stroke 1217 619 moveto (x) show 1408 715 moveto 224 /Times-Roman f1 (1) show 1786 380 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 1615 912 moveto (\266) show 1835 912 moveto 384 /Times-Italic f1 (x) show 2044 1008 moveto 224 /Times-Roman f1 (2) show 1592 520 moveto 622 0 rlineto stroke 2306 619 moveto 384 /Symbol f1 (-) show 2601 619 moveto 384 /Times-Italic f1 (x) show 2810 715 moveto 224 /Times-Roman f1 (2) show 3188 380 moveto 384 /Symbol f2 (\266) show 3035 912 moveto (\266) show 3255 912 moveto 384 /Times-Italic f1 (x) show 3446 1008 moveto 224 /Times-Roman f1 (1) show 3012 520 moveto 586 0 rlineto stroke 1024 355 moveto 384 /Symbol f1 (\346) show 1024 928 moveto (\350) show 1024 709 moveto (\347) show 3610 355 moveto (\366) show 3610 928 moveto (\370) show 3610 709 moveto (\367) show end pse gR gS 17 23 560 797 rC 72 294 :M f56 sf -.079(on a spinless particle in the state described by the wave function )A 378 282 66 18 rC 444 300 :M psb currentpoint pse 378 282 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2112 div 576 3 -1 roll exch div scale currentpoint translate 64 -1216 translate -4 1600 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 309 1696 moveto 224 /Times-Roman f1 (0) show 600 1600 moveto 384 /Times-Italic f1 (x) show 791 1696 moveto 224 /Times-Roman f1 (1) show 919 1600 moveto 384 /Times-Roman f1 (,) show 1054 1600 moveto 384 /Times-Italic f1 (x) show 1263 1696 moveto 224 /Times-Roman f1 (2) show 1409 1600 moveto 384 /Times-Roman f1 (,) show 1544 1600 moveto 384 /Times-Italic f1 (x) show 1750 1696 moveto 224 /Times-Roman f1 (3) show 459 1630 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1899 1630 moveto (\)) show end pse gR gS 17 23 560 797 rC 444 294 :M f56 sf .11 .011( are summarised)J 72 313 :M -.117(by the random variable, that we shall call )A 268 301 38 18 rC 306 319 :M psb currentpoint pse 268 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 576 3 -1 roll exch div scale currentpoint translate 64 -235 translate -4 619 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 715 moveto 224 /Times-Roman f1 (3) show 531 619 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 715 moveto 224 /Times-Roman f1 (0) show 388 649 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 649 moveto (\)) show end pse gR gS 17 23 560 797 rC 306 313 :M f56 sf -.116(, having characteristic function)A 194 338 202 23 rC 396 361 :M psb currentpoint pse 194 338 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6464 div 736 3 -1 roll exch div scale currentpoint translate 64 -2478 translate -4 2958 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) show 244 3091 moveto 224 /Times-Italic f1 (L) show 402 3147 moveto 160 /Times-Roman f1 (3) show 620 3091 moveto 224 /Symbol f2 (y) show 816 3147 moveto 160 /Times-Roman f1 (0) show 523 3109 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 224 1000 1383 /Symbol f3 (\() show 950 3109 moveto (\)) show 1200 2958 moveto 384 /Symbol f2 (t) show 1056 2968 moveto 384 1000 1171 /Symbol f3 (\() show 1419 2968 moveto (\)) show 1632 2958 moveto 384 /Symbol f1 (\272) show 1946 2958 moveto 384 /Times-Italic f1 (E) show 2326 2958 moveto (e) show 2515 2759 moveto 224 ns (i) show 2590 2759 moveto 224 /Symbol f1 (\327) show 2657 2759 moveto 224 /Symbol f2 (t) show 2789 2759 moveto 224 /Symbol f1 (\327) show 2856 2759 moveto 224 /Times-Italic f1 (L) show 3014 2815 moveto 160 /Times-Roman f1 (3) show 3232 2759 moveto 224 /Symbol f2 (y) show 3428 2815 moveto 160 /Times-Roman f1 (0) show 3135 2777 moveto 224 1000 1383 /Symbol f3 (\() show 3562 2777 moveto (\)) show 2187 3038 moveto 384 1000 1937 /Symbol f3 (\() show 3677 3038 moveto (\)) show 3891 2958 moveto 384 /Symbol f1 (=) show 4344 2958 moveto 384 /Symbol f2 (y) show 4657 3054 moveto 224 /Times-Roman f1 (0) show 4803 2958 moveto 384 /Times-Roman f1 (,) show 4936 2958 moveto 384 /Times-Italic f1 (e) show 5125 2777 moveto 224 ns (i) show 5200 2777 moveto 224 /Symbol f1 (\327) show 5267 2777 moveto 224 /Symbol f2 (t) show 5399 2777 moveto 224 /Symbol f1 (\327) show 5466 2777 moveto 224 /Times-Italic f1 (L) show 5624 2833 moveto 160 /Times-Roman f1 (3) show 5752 2958 moveto 384 /Symbol f2 (y) show 6065 3054 moveto 224 /Times-Roman f1 (0) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -91 266 4300 2593 vb 16 th 91 266 4209 2859 vb 16 th 91 266 6235 2593 vb 16 th -91 266 6326 2859 vb end pse gR gS 17 23 560 797 rC 396 353 :M f56 sf (.)S 72 394 :M -.123(If one introduces the Hamiltonian)A 72 428 :M ( )S 261 418 76 15 rC 337 433 :M psb currentpoint pse 261 418 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2432 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -7 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 285 357 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (w) show 478 357 moveto 224 /Times-Roman f1 (,) show 551 357 moveto 224 /Symbol f2 (e) show 808 261 moveto 384 /Symbol f1 (=) show 1123 261 moveto 384 /Times-Italic f1 (h) show 1319 357 moveto 224 /Times-Roman f1 (1) show 1425 357 moveto 224 /Times-Roman f1 (,) show 1496 357 moveto 224 /Times-Roman f1 (2) show 1727 261 moveto 384 /Symbol f1 (+) show 2026 261 moveto 384 /Times-Italic f1 (h) show 2237 357 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 337 428 :M f56 sf ( \(6\))S 72 448 :M -.077(where)A 205 452 184 28 rC 389 480 :M psb currentpoint pse 205 452 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5888 div 896 3 -1 roll exch div scale currentpoint translate 64 -1228 translate -3 1772 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 193 1868 moveto 224 /Times-Roman f1 (1) show 299 1868 moveto 224 /Times-Roman f1 (,) show 370 1868 moveto 224 /Times-Roman f1 (2) show 616 1772 moveto 384 /Symbol f1 (=) show 955 1533 moveto 384 /Times-Roman f1 (1) show 955 2065 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 934 1673 moveto 235 0 rlineto stroke 1233 1772 moveto 384 /Times-Italic f1 (p) show 1421 1868 moveto 224 /Times-Roman f1 (1) show 1535 1606 moveto (2) show 1766 1772 moveto 384 /Symbol f1 (+) show 2089 1533 moveto 384 /Times-Roman f1 (1) show 2089 2065 moveto (2) show 2068 1673 moveto 235 0 rlineto stroke 2367 1772 moveto 384 /Times-Italic f1 (p) show 2573 1868 moveto 224 /Times-Roman f1 (2) show 2705 1606 moveto (2) show 2936 1772 moveto 384 /Symbol f1 (+) show 3259 1533 moveto 384 /Times-Roman f1 (1) show 3259 2065 moveto (2) show 3238 1673 moveto 235 0 rlineto stroke 3521 1772 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (w) show 3847 1601 moveto 224 /Times-Roman f1 (2) show 4055 1772 moveto 384 /Symbol f1 (\327) show 4333 1772 moveto 384 /Times-Italic f1 (q) show 4532 1868 moveto 224 /Times-Roman f1 (1) show 4646 1606 moveto (2) show 4877 1772 moveto 384 /Symbol f1 (+) show 5170 1772 moveto 384 /Times-Italic f1 (q) show 5387 1868 moveto 224 /Times-Roman f1 (2) show 5519 1606 moveto (2) show 4195 1814 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1515 /Symbol f3 (\() show 5678 1814 moveto (\)) show end pse gR gS 212 499 170 28 rC 382 527 :M psb currentpoint pse 212 499 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5440 div 896 3 -1 roll exch div scale currentpoint translate 64 -577 translate -3 1121 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 208 1217 moveto 224 /Times-Roman f1 (3) show 444 1121 moveto 384 /Symbol f1 (=) show 783 882 moveto 384 /Times-Roman f1 (1) show 783 1414 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 762 1022 moveto 235 0 rlineto stroke 1061 1121 moveto 384 /Times-Italic f1 (p) show 1264 1217 moveto 224 /Times-Roman f1 (3) show 1386 955 moveto (2) show 1616 1121 moveto 384 /Symbol f1 (-) show 1909 1121 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 2180 1121 moveto 384 /Symbol f1 (\327) show 2331 1121 moveto 384 /Times-Italic f1 (q) show 2545 1217 moveto 224 /Times-Roman f1 (3) show 2743 1121 moveto 384 /Symbol f1 (\327) show 3021 1121 moveto 384 /Times-Italic f1 (q) show 3220 1217 moveto 224 /Times-Roman f1 (1) show 3410 1121 moveto 384 /Symbol f1 (\327) show 3570 1121 moveto 384 /Times-Italic f1 (p) show 3776 1217 moveto 224 /Times-Roman f1 (2) show 4006 1121 moveto 384 /Symbol f1 (-) show 4298 1121 moveto 384 /Times-Italic f1 (q) show 4515 1217 moveto 224 /Times-Roman f1 (2) show 4723 1121 moveto 384 /Symbol f1 (\327) show 4883 1121 moveto 384 /Times-Italic f1 (p) show 5071 1217 moveto 224 /Times-Roman f1 (1) show 2883 1149 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1369 /Symbol f3 (\() show 5212 1149 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 542 :M f56 sf -.163(and)A 237 546 110 33 rC 347 579 :M psb currentpoint pse 237 546 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3520 div 1056 3 -1 roll exch div scale currentpoint translate 64 51 translate 0 525 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 224 621 moveto 224 ns (j) show 411 525 moveto 384 /Symbol f1 (=) show 719 286 moveto 384 /Times-Roman f1 (1) show 762 818 moveto 384 /Times-Italic f1 (i) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 729 426 moveto 173 0 rlineto stroke 1139 286 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 989 818 moveto (\266) show 1209 818 moveto 384 /Times-Italic f1 (x) show 1436 914 moveto 224 ns (j) show 966 426 moveto 581 0 rlineto stroke 1587 525 moveto 384 /Times-Roman f1 (,) show 1728 525 moveto 384 /Times-Roman f1 ( ) show 1920 525 moveto 384 /Times-Italic f1 (j) show 2102 525 moveto 384 /Times-Roman f1 (=) show 2377 525 moveto 384 /Times-Roman f1 (1) show 2536 525 moveto 384 /Times-Roman f1 (,) show 2677 525 moveto ( ) show 2794 525 moveto 384 /Times-Roman f1 (2) show 2984 525 moveto 384 /Times-Roman f1 (,) show 3125 525 moveto ( ) show 3237 525 moveto 384 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 347 564 :M f56 sf .455 .045( ,)J 72 612 :M -.046(and sets)A 256 634 78 18 rC 334 652 :M psb currentpoint pse 256 634 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2496 div 576 3 -1 roll exch div scale currentpoint translate 64 -4629 translate -4 5045 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 5141 moveto 224 /Times-Italic f1 (t) show 514 5045 moveto 384 /Symbol f1 (=) show 824 5045 moveto 384 /Times-Italic f1 (e) show 1015 4844 moveto 224 /Symbol f1 (-) show 1162 4844 moveto 224 /Times-Italic f1 (i) show 1237 4844 moveto 224 /Symbol f1 (\327) show 1304 4844 moveto 224 /Times-Italic f1 (t) show 1383 4844 moveto 224 /Symbol f1 (\327) show 1448 4844 moveto 224 /Times-Italic f1 (H) show 1631 4900 moveto 160 /Symbol f2 (w) show 1762 4900 moveto 160 /Times-Roman f1 (,) show 1807 4900 moveto 160 /Symbol f2 (e) show 1951 5045 moveto 384 ns (y) show 2264 5141 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 335 647 :M f56 sf (,)S 72 685 :M -.182(it is elementary to check that)A 72 728 :M ( )S 189 707 229 36 rC 418 743 :M psb currentpoint pse 189 707 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7328 div 1152 3 -1 roll exch div scale currentpoint translate 32 -4082 translate 167 4754 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 480 4850 moveto 224 /Times-Roman f1 (0) show 626 4754 moveto 384 /Times-Roman f1 (,) show 759 4754 moveto 384 /Times-Italic f1 (e) show 948 4573 moveto 224 ns (i) show 1023 4573 moveto 224 /Symbol f1 (\327) show 1090 4573 moveto 224 /Symbol f2 (t) show 1222 4573 moveto 224 /Symbol f1 (\327) show 1289 4573 moveto 224 /Times-Italic f1 (L) show 1447 4629 moveto 160 /Times-Roman f1 (3) show 1575 4754 moveto 384 /Symbol f2 (y) show 1888 4850 moveto 224 /Times-Roman f1 (0) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -91 266 123 4389 vb 16 th 91 266 32 4655 vb 16 th 91 266 2058 4389 vb 16 th -91 266 2149 4655 vb 2273 4754 moveto 384 /Symbol f1 (=) show 2584 4754 moveto 384 /Times-Roman f1 (lim) show 2600 4978 moveto 224 /Times-Italic f1 (t) show 2684 4978 moveto 224 /Symbol f1 (\256) show 2925 4978 moveto (\245) show 3311 4754 moveto 384 /Symbol f2 (y) show 3627 4850 moveto 224 /Times-Italic f1 (t) show 3729 4754 moveto 384 /Times-Roman f1 (,) show 3862 4754 moveto (exp) show 4612 4754 moveto 384 /Times-Italic f1 (i) show 4781 4754 moveto 384 /Symbol f1 (\327) show 4938 4754 moveto 384 /Symbol f2 (t) show 5206 4754 moveto 384 /Symbol f1 (\327) show 5730 4515 moveto 384 /Times-Italic f1 (q) show 5944 4611 moveto 224 /Times-Roman f1 (3) show 5390 5047 moveto 384 /Symbol f2 (e) show 5661 5047 moveto 384 /Symbol f1 (\327) show 5817 5047 moveto 384 /Times-Italic f1 (t) show 5950 4876 moveto 224 /Times-Roman f1 (2) show 6231 5047 moveto 384 ns (2) show 8 th -130 406 6224 4724 vb 16 th 5366 4655 moveto 1079 0 rlineto stroke 4421 4503 moveto 384 /Symbol f1 (\346) show 4421 5050 moveto (\350) show 4421 4831 moveto (\347) show 6457 4503 moveto (\366) show 6457 5050 moveto (\370) show 6457 4831 moveto (\367) show 6657 4754 moveto 384 /Symbol f2 (y) show 6973 4850 moveto 224 /Times-Italic f1 (t) show -146 539 3267 4116 vb 16 th 146 539 3121 4655 vb 16 th 146 539 7099 4116 vb 16 th -146 539 7245 4655 vb end pse gR gS 17 23 560 797 rC 418 728 :M f56 sf ( \(7\))S endp showpage %%Page: 10 10 %%BeginPageSetup initializepage (defalco; pag: 10 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (10)S 72 94 :M -.097(The proof of this statement requires only the expansion)A 170 116 255 36 rC 425 152 :M psb currentpoint pse 170 116 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8160 div 1152 3 -1 roll exch div scale currentpoint translate 64 -5370 translate -4 6042 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 309 6138 moveto 224 /Times-Roman f1 (0) show 600 6042 moveto 384 /Times-Italic f1 (x) show 791 6138 moveto 224 /Times-Roman f1 (1) show 919 6042 moveto 384 /Times-Roman f1 (,) show 1054 6042 moveto 384 /Times-Italic f1 (x) show 1263 6138 moveto 224 /Times-Roman f1 (2) show 1409 6042 moveto 384 /Times-Roman f1 (,) show 1544 6042 moveto 384 /Times-Italic f1 (x) show 1750 6138 moveto 224 /Times-Roman f1 (3) show 459 6072 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1899 6072 moveto (\)) show 2113 6042 moveto 384 /Symbol f1 (=) show 3069 5751 moveto 384 /Symbol f2 (j) show 3341 5847 moveto 224 /Times-Italic f1 (n) show 3465 5903 moveto 160 /Times-Roman f1 (1) show 3578 5847 moveto 224 /Times-Roman f1 (,) show 3653 5847 moveto 224 /Times-Italic f1 (n) show 3790 5903 moveto 160 /Times-Roman f1 (2) show 4083 5751 moveto 384 /Times-Italic f1 (x) show 4274 5847 moveto 224 /Times-Roman f1 (1) show 4402 5751 moveto 384 /Times-Roman f1 (,) show 4537 5751 moveto 384 /Times-Italic f1 (x) show 4746 5847 moveto 224 /Times-Roman f1 (2) show 3942 5779 moveto 384 1000 1369 /Symbol f3 (\() show 4905 5779 moveto (\)) show 3831 6335 moveto 384 /Times-Roman f1 (2) show 4083 6335 moveto 384 /Symbol f1 (\327) show 4240 6335 moveto 384 /Symbol f2 (p) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 932 380 384 3580 6374 16 sqr /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3050 5943 moveto 1993 0 rlineto stroke 5402 6042 moveto (g) show 5635 6138 moveto 224 /Times-Italic f1 (n) show 5759 6194 moveto 160 /Times-Roman f1 (1) show 5872 6138 moveto 224 /Times-Roman f1 (,) show 5947 6138 moveto 224 /Times-Italic f1 (n) show 6084 6194 moveto 160 /Times-Roman f1 (2) show 6383 6042 moveto 384 /Times-Italic f1 (p) show 6236 6063 moveto 384 1000 1291 /Symbol f3 (\() show 6584 6063 moveto (\)) show 6760 6042 moveto 384 /Symbol f1 (\327) show 6912 6042 moveto 384 /Times-Italic f1 (e) show 7101 5861 moveto 224 ns (i) show 7176 5861 moveto 224 /Symbol f1 (\327) show 7245 5861 moveto 224 /Times-Italic f1 (p) show 7367 5861 moveto 224 /Symbol f1 (\327) show 7433 5861 moveto 224 /Times-Italic f1 (x) show 7565 5917 moveto 160 /Times-Roman f1 (3) show 7688 6042 moveto 384 /Times-Italic f1 (dp) show 5105 6461 moveto 224 /Symbol f1 (-) show 5251 6461 moveto (\245) show 5105 5593 moveto (+) show 5252 5593 moveto (\245) show 5172 6176 moveto 576 ns (\362) show 2429 6415 moveto 224 /Times-Italic f1 (n) show 2553 6471 moveto 160 /Times-Roman f1 (1) show 2666 6415 moveto 224 /Times-Roman f1 (,) show 2741 6415 moveto 224 /Times-Italic f1 (n) show 2878 6471 moveto 160 /Times-Roman f1 (2) show 2504 6129 moveto 576 /Symbol f1 (\345) show end pse gR gS 17 23 560 797 rC 72 184 :M f56 sf 2.504 .25(of the initial condition )J 198 174 16 15 rC 214 189 :M psb currentpoint pse 198 174 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 480 3 -1 roll exch div scale currentpoint translate 64 -5722 translate -4 6042 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 309 6138 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 214 184 :M f56 sf 3.199 .32( in terms of a complete orthonormal set )J 444 171 69 30 rC 513 201 :M psb currentpoint pse 444 171 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2208 div 960 3 -1 roll exch div scale currentpoint translate 64 -7352 translate 149 7768 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 421 7864 moveto 224 /Times-Italic f1 (n) show 545 7920 moveto 160 /Times-Roman f1 (1) show 658 7864 moveto 224 /Times-Roman f1 (,) show 733 7864 moveto 224 /Times-Italic f1 (n) show 870 7920 moveto 160 /Times-Roman f1 (2) show -22 7831 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1723 /Symbol f3 ({) show 1012 7831 moveto (}) show 1202 7970 moveto 224 /Times-Italic f1 (n) show 1326 8026 moveto 160 /Times-Roman f1 (1) show 1451 7970 moveto 224 /Symbol f1 (=) show 1598 7970 moveto 224 /Times-Roman f1 (0) show 1722 7970 moveto 224 /Times-Roman f1 (,) show 1775 7970 moveto 224 /Times-Roman f1 (1) show 1881 7970 moveto 224 /Times-Roman f1 (,) show 1943 7970 moveto (.) show 2000 7970 moveto (.) show 2057 7970 moveto (.) show 1202 8202 moveto 224 /Times-Italic f1 (n) show 1339 8258 moveto 160 /Times-Roman f1 (2) show 1477 8202 moveto 224 /Symbol f1 (=) show 1624 8202 moveto 224 /Times-Roman f1 (0) show 1748 8202 moveto 224 /Times-Roman f1 (,) show 1801 8202 moveto 224 /Times-Roman f1 (1) show 1907 8202 moveto 224 /Times-Roman f1 (,) show 1969 8202 moveto (.) show 2026 8202 moveto (.) show end pse gR gS 17 23 560 797 rC 513 184 :M f56 sf (of)S 72 214 :M -.099(simultaneous eigenfunctions of )A 222 204 18 15 rC 240 219 :M psb currentpoint pse 222 204 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 576 div 480 3 -1 roll exch div scale currentpoint translate 64 -1374 translate -3 1694 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 193 1790 moveto 224 /Times-Roman f1 (1) show 299 1790 moveto 224 /Times-Roman f1 (,) show 370 1790 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 240 214 :M f56 sf -.081( and )A 263 204 14 15 rC 277 219 :M psb currentpoint pse 263 204 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 -4057 translate -4 4377 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 4473 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 277 214 :M f56 sf -.066( satisfying)A 213 238 169 18 rC 382 256 :M psb currentpoint pse 213 238 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5408 div 576 3 -1 roll exch div scale currentpoint translate 64 59 translate -3 325 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 226 421 moveto 224 /Times-Roman f1 (1) show 332 421 moveto 224 /Times-Roman f1 (,) show 403 421 moveto 224 /Times-Roman f1 (2) show 528 325 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 800 421 moveto 224 /Times-Italic f1 (n) show 937 477 moveto 160 /Times-Roman f1 (1) show 1050 421 moveto 224 /Times-Roman f1 (,) show 1125 421 moveto 224 /Times-Italic f1 (n) show 1275 477 moveto 160 /Times-Roman f1 (2) show 1523 325 moveto 384 /Symbol f1 (=) show 1965 325 moveto 384 /Times-Italic f1 (n) show 2183 421 moveto 224 /Times-Roman f1 (1) show 2396 325 moveto 384 /Symbol f1 (+) show 2695 325 moveto 384 /Times-Italic f1 (n) show 2931 421 moveto 224 /Times-Roman f1 (2) show 3162 325 moveto 384 /Symbol f1 (+) show 3422 325 moveto 384 /Times-Roman f1 (1) show 1821 353 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1369 /Symbol f3 (\() show 3594 353 moveto (\)) show 3770 325 moveto 384 /Symbol f1 (\327) show 3914 325 moveto 384 /Symbol f2 (w) show 4284 325 moveto 384 /Symbol f1 (\327) show 4431 325 moveto 384 /Symbol f2 (j) show 4703 421 moveto 224 /Times-Italic f1 (n) show 4840 477 moveto 160 /Times-Roman f1 (1) show 4953 421 moveto 224 /Times-Roman f1 (,) show 5028 421 moveto 224 /Times-Italic f1 (n) show 5178 477 moveto 160 /Times-Roman f1 (2) show end pse gR gS 231 257 129 18 rC 360 275 :M psb currentpoint pse 231 257 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4128 div 576 3 -1 roll exch div scale currentpoint translate 64 -9023 translate -4 9407 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 9503 moveto 224 /Times-Roman f1 (3) show 363 9407 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 635 9503 moveto 224 /Times-Italic f1 (n) show 759 9559 moveto 160 /Times-Roman f1 (1) show 872 9503 moveto 224 /Times-Roman f1 (,) show 947 9503 moveto 224 /Times-Italic f1 (n) show 1084 9559 moveto 160 /Times-Roman f1 (2) show 1332 9407 moveto 384 /Symbol f1 (=) show 1774 9407 moveto 384 /Times-Italic f1 (n) show 1970 9503 moveto 224 /Times-Roman f1 (1) show 2182 9407 moveto 384 /Symbol f1 (-) show 2480 9407 moveto 384 /Times-Italic f1 (n) show 2694 9503 moveto 224 /Times-Roman f1 (2) show 1630 9435 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1369 /Symbol f3 (\() show 2853 9435 moveto (\)) show 3029 9407 moveto 384 /Symbol f1 (\327) show 3176 9407 moveto 384 /Symbol f2 (j) show 3448 9503 moveto 224 /Times-Italic f1 (n) show 3572 9559 moveto 160 /Times-Roman f1 (1) show 3685 9503 moveto 224 /Times-Roman f1 (,) show 3760 9503 moveto 224 /Times-Italic f1 (n) show 3897 9559 moveto 160 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 360 269 :M f56 sf (.)S 72 307 :M 1.976 .198(The observation that, on each component of angular momentum )J 412 297 35 14 rC 447 311 :M psb currentpoint pse 412 297 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 448 3 -1 roll exch div scale currentpoint translate -1713 -9087 translate 1774 9407 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (n) show 1970 9503 moveto 224 /Times-Roman f1 (1) show 2182 9407 moveto 384 /Symbol f1 (-) show 2480 9407 moveto 384 /Times-Italic f1 (n) show 2694 9503 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 447 307 :M f56 sf 2.062 .206(, the evolution)J 72 324 :M .737 .074(according to )J 137 314 24 15 rC 161 329 :M psb currentpoint pse 137 314 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 768 div 480 3 -1 roll exch div scale currentpoint translate 64 -2399 translate -7 2719 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 285 2815 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (w) show 478 2815 moveto 224 /Times-Roman f1 (,) show 551 2815 moveto 224 /Symbol f2 (e) show end pse gR gS 17 23 560 797 rC 161 324 :M f56 sf 1.216 .122( reduces to the easily solvable problem of one-dimensional motion in a)J 72 342 :M -.131(constant force field )A 165 330 93 18 rC 258 348 :M psb currentpoint pse 165 330 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2976 div 576 3 -1 roll exch div scale currentpoint translate 64 -9023 translate -7 9407 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (f) show 122 9503 moveto 224 ns (n) show 246 9559 moveto 160 /Times-Roman f1 (1) show 359 9503 moveto 224 /Times-Roman f1 (,) show 434 9503 moveto 224 /Times-Italic f1 (n) show 571 9559 moveto 160 /Times-Roman f1 (2) show 819 9407 moveto 384 /Symbol f1 (=) show 1129 9407 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 1400 9407 moveto 384 /Symbol f1 (\327) show 1684 9407 moveto 384 /Times-Italic f1 (n) show 1880 9503 moveto 224 /Times-Roman f1 (1) show 2092 9407 moveto 384 /Symbol f1 (-) show 2390 9407 moveto 384 /Times-Italic f1 (n) show 2604 9503 moveto 224 /Times-Roman f1 (2) show 1540 9435 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1369 /Symbol f3 (\() show 2763 9435 moveto (\)) show end pse gR gS 17 23 560 797 rC 258 342 :M f56 sf -.054( gives, then, the explicit expression)A 226 367 143 36 rC 369 403 :M psb currentpoint pse 226 367 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4576 div 1152 3 -1 roll exch div scale currentpoint translate 64 34 translate 190 638 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 506 734 moveto 224 /Times-Italic f1 (t) show 608 638 moveto 384 /Times-Roman f1 (,) show 741 638 moveto (exp) show 1491 638 moveto 384 /Times-Italic f1 (i) show 1660 638 moveto 384 /Symbol f1 (\327) show 1817 638 moveto 384 /Symbol f2 (t) show 2085 638 moveto 384 /Symbol f1 (\327) show 2609 399 moveto 384 /Times-Italic f1 (q) show 2823 495 moveto 224 /Times-Roman f1 (3) show 2269 931 moveto 384 /Symbol f2 (e) show 2540 931 moveto 384 /Symbol f1 (\327) show 2696 931 moveto 384 /Times-Italic f1 (t) show 2829 760 moveto 224 /Times-Roman f1 (2) show 3110 931 moveto 384 ns (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 3103 608 vb 16 th 2245 539 moveto 1079 0 rlineto stroke 1300 387 moveto 384 /Symbol f1 (\346) show 1300 934 moveto (\350) show 1300 715 moveto (\347) show 3336 387 moveto (\366) show 3336 934 moveto (\370) show 3336 715 moveto (\367) show 3536 638 moveto 384 /Symbol f2 (y) show 3852 734 moveto 224 /Times-Italic f1 (t) show -146 539 146 0 vb 16 th 146 539 0 539 vb 16 th 146 539 3978 0 vb 16 th -146 539 4124 539 vb 4248 638 moveto 384 /Symbol f1 (=) show end pse gR gS 17 23 560 797 rC 72 429 :M f56 sf ( )S 156 404 305 40 rC 461 444 :M psb currentpoint pse 156 404 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9760 div 1280 3 -1 roll exch div scale currentpoint translate 64 -1253 translate -4 2053 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (=) show 843 2053 moveto 384 /Times-Italic f1 (e) show 1032 1856 moveto 224 ns (i) show 1107 1856 moveto 224 /Symbol f1 (\327) show 1174 1856 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (t) show 1306 1856 moveto 224 /Symbol f1 (\327) show 1461 1856 moveto 224 /Times-Italic f1 (n) show 1585 1912 moveto 160 /Times-Roman f1 (1) show 1710 1856 moveto 224 /Symbol f1 (-) show 1859 1856 moveto 224 /Times-Italic f1 (n) show 1996 1912 moveto 160 /Times-Roman f1 (2) show 1364 1873 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 224 1000 1361 /Symbol f3 (\() show 2130 1873 moveto (\)) show 312 2426 moveto 224 /Times-Italic f1 (n) show 436 2482 moveto 160 /Times-Roman f1 (1) show 549 2426 moveto 224 /Times-Roman f1 (,) show 624 2426 moveto 224 /Times-Italic f1 (n) show 761 2482 moveto 160 /Times-Roman f1 (2) show 387 2140 moveto 576 /Symbol f1 (\345) show 2294 2053 moveto 384 ns (\327) show 2815 2053 moveto 384 /Symbol f2 (g) show 3048 2149 moveto 224 /Times-Italic f1 (n) show 3172 2205 moveto 160 /Times-Roman f1 (1) show 3285 2149 moveto 224 /Times-Roman f1 (,) show 3360 2149 moveto 224 /Times-Italic f1 (n) show 3497 2205 moveto 160 /Times-Roman f1 (2) show 3796 2053 moveto 384 /Times-Italic f1 (p) show 3649 2074 moveto 384 1000 1291 /Symbol f3 (\() show 3997 2074 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2816 1685 moveto 1287 0 rlineto stroke 4173 2053 moveto 384 /Symbol f1 (\327) show 4332 2053 moveto 384 /Symbol f2 (g) show 4565 2149 moveto 224 /Times-Italic f1 (n) show 4689 2205 moveto 160 /Times-Roman f1 (1) show 4802 2149 moveto 224 /Times-Roman f1 (,) show 4877 2149 moveto 224 /Times-Italic f1 (n) show 5014 2205 moveto 160 /Times-Roman f1 (2) show 5370 2053 moveto 384 /Times-Italic f1 (p) show 5642 2053 moveto 384 /Symbol f1 (-) show 6382 1814 moveto 384 /Symbol f2 (t) show 5967 2346 moveto (e) show 6238 2346 moveto 384 /Symbol f1 (\327) show 6394 2346 moveto 384 /Times-Italic f1 (t) show 6527 2175 moveto 224 /Times-Roman f1 (2) show 6808 2346 moveto 384 ns (2) show 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 6801 2023 vb 16 th 5943 1954 moveto 1079 0 rlineto stroke 5171 1802 moveto 384 /Symbol f1 (\346) show 5171 2349 moveto (\350) show 5171 2130 moveto (\347) show 7034 1802 moveto (\366) show 7034 2349 moveto (\370) show 7034 2130 moveto (\367) show 7276 2053 moveto (\327) show 7428 2053 moveto 384 /Times-Italic f1 (e) show 7615 1677 moveto 224 /Times-Roman f1 (2) show 7738 1677 moveto 224 /Symbol f1 (\327) show 7803 1677 moveto 224 /Times-Italic f1 (i) show 7878 1677 moveto 224 /Symbol f1 (\327) show 8086 1532 moveto 224 /Times-Italic f1 (p) show 8208 1532 moveto 224 /Symbol f1 (\327) show 8275 1532 moveto 224 /Symbol f2 (t) show 8105 1852 moveto (e) show 8239 1852 moveto 224 /Symbol f1 (\327) show 8306 1852 moveto 224 /Times-Italic f1 (t) show 8 th 8068 1620 moveto 340 0 rlineto stroke 8438 1677 moveto 224 /Symbol f1 (-) show 8741 1535 moveto 224 /Symbol f2 (t) show 8884 1436 moveto 160 /Times-Roman f1 (2) show 8603 1867 moveto 224 /Symbol f2 (e) show 8748 1768 moveto 160 /Times-Roman f1 (2) show 8873 1867 moveto 224 /Symbol f1 (\327) show 8940 1867 moveto 224 /Times-Italic f1 (t) show 9027 1768 moveto 160 /Times-Roman f1 (3) show 8589 1620 moveto 556 0 rlineto stroke 7939 1517 moveto 224 /Symbol f1 (\346) show 7939 1865 moveto (\350) show 7939 1736 moveto (\347) show 9165 1517 moveto (\366) show 9165 1865 moveto (\370) show 9165 1736 moveto (\367) show 9288 2053 moveto 384 /Times-Italic f1 (dp) show 2452 2472 moveto 224 /Symbol f1 (-) show 2598 2472 moveto (\245) show 2452 1604 moveto (+) show 2599 1604 moveto (\245) show 2519 2187 moveto 576 ns (\362) show end pse gR gS 17 23 560 797 rC 461 429 :M f56 sf ( \(8\))S 72 477 :M -.124(from which the limit relation \(7\) immediately follows.)A 72 513 :M -.119(It is instructive to consider the particular case of a Gaussian initial distribution)A 72 557 :M ( )S 186 535 220 41 rC 406 576 :M psb currentpoint pse 186 535 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7040 div 1312 3 -1 roll exch div scale currentpoint translate 64 37 translate -4 667 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 309 763 moveto 224 /Times-Roman f1 (0) show 600 667 moveto 384 /Times-Italic f1 (x) show 791 763 moveto 224 /Times-Roman f1 (1) show 919 667 moveto 384 /Times-Roman f1 (,) show 1054 667 moveto 384 /Times-Italic f1 (x) show 1263 763 moveto 224 /Times-Roman f1 (2) show 1409 667 moveto 384 /Times-Roman f1 (,) show 1544 667 moveto 384 /Times-Italic f1 (x) show 1750 763 moveto 224 /Times-Roman f1 (3) show 459 697 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1899 697 moveto (\)) show 2113 667 moveto 384 /Symbol f1 (=) show 3783 428 moveto 384 /Times-Roman f1 (1) show 3091 1038 moveto (2) show 3343 1038 moveto 384 /Symbol f1 (\327) show 3500 1038 moveto 384 /Symbol f2 (p) show 3810 1038 moveto 384 /Symbol f1 (\327) show 3955 1038 moveto 384 /Symbol f2 (a) show 4255 1134 moveto 224 /Times-Italic f1 (j) show 2955 1093 moveto 384 1000 1661 /Symbol f3 (\() show 4355 1093 moveto (\)) show 4469 760 moveto 224 /Times-Roman f1 (1) show 4646 760 moveto (4) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -64 200 4623 608 vb 16 th 2943 568 moveto 1872 0 rlineto stroke 2515 1037 moveto 224 /Times-Italic f1 (j) show 2592 1037 moveto 224 /Symbol f1 (=) show 2720 1037 moveto 224 /Times-Roman f1 (1) show 2603 267 moveto (3) show 2417 755 moveto 576 /Symbol f1 (\325) show 4871 667 moveto 384 /Times-Roman f1 (exp) show 5625 667 moveto 384 /Symbol f1 (-) show 6089 384 moveto 384 /Times-Italic f1 (x) show 6316 480 moveto 224 ns (j) show 6389 218 moveto 224 /Times-Roman f1 (2) show 5922 960 moveto 384 ns (4) show 6173 960 moveto 384 /Symbol f1 (\327) show 6318 960 moveto 384 /Symbol f2 (a) show 6618 1056 moveto 224 /Times-Italic f1 (j) show 5894 568 moveto 835 0 rlineto stroke 5430 355 moveto 384 /Symbol f1 (\346) show 5430 1024 moveto (\350) show 5430 805 moveto (\347) show 6741 355 moveto (\366) show 6741 1024 moveto (\370) show 6741 805 moveto (\367) show end pse gR gS 17 23 560 797 rC 406 557 :M f56 sf ( \(9\))S 72 609 :M -.089(\(all numerical examples in this paper will refer to this particular case; moreover, we shall set, in)A 72 627 :M .208 .021(what follows, )J 141 617 29 11 rC 170 628 :M psb currentpoint pse 141 617 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 928 div 352 3 -1 roll exch div scale currentpoint translate 64 59 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show 397 261 moveto 384 /Symbol f1 (=) show 673 261 moveto 384 /Times-Roman f1 (1) show end pse gR gS 17 23 560 797 rC 170 627 :M f56 sf .909 .091(, )J 177 617 26 12 rC 177 629 :M f167 sf .269 .027( )J 203 629 :M psb currentpoint pse 177 617 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 384 3 -1 roll exch div scale currentpoint translate 64 -518 translate -8 838 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /MT-Extra f1 (h) show 281 838 moveto 384 /Symbol f1 (=) show 557 838 moveto 384 /Times-Roman f1 (1) show end pse gR gS 17 23 560 797 rC 203 627 :M f56 sf -.105(\). The right hand side of \(8\) can be explicitly computed in this case:)A 72 670 :M ( )S 144 649 315 36 rC 459 685 :M psb currentpoint pse 144 649 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10080 div 1152 3 -1 roll exch div scale currentpoint translate 64 34 translate 190 638 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 506 734 moveto 224 /Times-Italic f1 (t) show 625 638 moveto 384 /Times-Roman f1 (,) show 758 638 moveto (exp) show 1508 638 moveto 384 /Times-Italic f1 (i) show 1711 638 moveto 384 /Symbol f1 (\327) show 1868 638 moveto 384 /Symbol f2 (t) show 2136 638 moveto 384 /Symbol f1 (\327) show 2663 399 moveto 384 /Times-Italic f1 (q) show 2899 495 moveto 224 /Times-Roman f1 (3) show 2320 931 moveto 384 /Symbol f2 (e) show 2591 931 moveto 384 /Symbol f1 (\327) show 2747 931 moveto 384 /Times-Italic f1 (t) show 2908 760 moveto 224 /Times-Roman f1 (2) show 3189 931 moveto 384 ns (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 3182 608 vb 16 th 2296 539 moveto 1107 0 rlineto stroke 1317 387 moveto 384 /Symbol f1 (\346) show 1317 934 moveto (\350) show 1317 715 moveto (\347) show 3415 387 moveto (\366) show 3415 934 moveto (\370) show 3415 715 moveto (\367) show 3615 638 moveto 384 /Symbol f2 (y) show 3931 734 moveto 224 /Times-Italic f1 (t) show -146 539 146 0 vb 16 th 146 539 0 539 vb 16 th 146 539 4074 0 vb 16 th -146 539 4220 539 vb 4344 638 moveto 384 /Symbol f1 (=) show 4658 638 moveto 384 /Symbol f2 (c) show 4906 771 moveto 224 /Times-Italic f1 (L) show 5084 827 moveto 160 /Times-Roman f1 (3) show 5302 771 moveto 224 /Symbol f2 (y) show 5498 827 moveto 160 /Times-Roman f1 (0) show 5205 789 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 224 1000 1383 /Symbol f3 (\() show 5632 789 moveto (\)) show 5882 638 moveto 384 /Symbol f2 (t) show 5738 648 moveto 384 1000 1171 /Symbol f3 (\() show 6101 648 moveto (\)) show 6277 638 moveto 384 /Symbol f1 (\327) show 6429 638 moveto 384 /Times-Roman f1 (exp) show 7183 638 moveto 384 /Symbol f1 (-) show 7481 399 moveto 384 /Symbol f2 (t) show 7705 228 moveto 224 /Times-Roman f1 (2) show 7567 931 moveto 384 ns (2) show 16 th 7452 539 moveto 423 0 rlineto stroke 7945 638 moveto 384 /Symbol f1 (\327) show 8133 399 moveto 384 /Times-Roman f1 (4) show 8384 399 moveto 384 /Symbol f1 (\327) show 8529 399 moveto 384 /Symbol f2 (a) show 8808 495 moveto 224 /Times-Roman f1 (3) show 8930 233 moveto (2) show 9161 399 moveto 384 /Symbol f1 (+) show 9459 399 moveto 384 /Times-Italic f1 (t) show 9620 228 moveto 224 /Times-Roman f1 (2) show 8186 931 moveto 384 /Symbol f2 (e) show 8413 760 moveto 224 /Times-Roman f1 (2) show 8621 931 moveto 384 /Symbol f1 (\327) show 8766 931 moveto 384 /Symbol f2 (a) show 9045 1027 moveto 224 /Times-Roman f1 (3) show 9243 931 moveto 384 /Symbol f1 (\327) show 9399 931 moveto 384 /Times-Italic f1 (t) show 9564 760 moveto 224 /Times-Roman f1 (4) show 8105 539 moveto 1685 0 rlineto stroke 6988 371 moveto 384 /Symbol f1 (\346) show 6988 950 moveto (\350) show 6988 731 moveto (\347) show 9802 371 moveto (\366) show 9802 950 moveto (\370) show 9802 731 moveto (\367) show end pse gR gS 17 23 560 797 rC 459 670 :M f56 sf ( \(10\))S 72 718 :M -.077(where)A endp showpage %%Page: 11 11 %%BeginPageSetup initializepage (defalco; pag: 11 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (11)S 213 86 169 53 rC 382 139 :M psb currentpoint pse 213 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5408 div 1696 3 -1 roll exch div scale currentpoint translate 64 -2712 translate -4 3256 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (c) show 244 3389 moveto 224 /Times-Italic f1 (L) show 402 3445 moveto 160 /Times-Roman f1 (3) show 620 3389 moveto 224 /Symbol f2 (y) show 816 3445 moveto 160 /Times-Roman f1 (0) show 523 3407 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 224 1000 1383 /Symbol f3 (\() show 950 3407 moveto (\)) show 1200 3256 moveto 384 /Symbol f2 (t) show 1056 3266 moveto 384 1000 1171 /Symbol f3 (\() show 1419 3266 moveto (\)) show 1633 3256 moveto 384 /Symbol f1 (=) show 3530 3017 moveto 384 /Times-Roman f1 (1) show 2203 3925 moveto (1) show 2447 3925 moveto 384 /Symbol f1 (+) show 3313 3659 moveto 384 /Symbol f2 (a) show 3577 3755 moveto 224 /Times-Roman f1 (1) show 3789 3659 moveto 384 /Symbol f1 (-) show 4075 3659 moveto 384 /Symbol f2 (a) show 4357 3755 moveto 224 /Times-Roman f1 (2) show 3181 3687 moveto 384 1000 1369 /Symbol f3 (\() show 4516 3687 moveto (\)) show 4648 3429 moveto 224 /Times-Roman f1 (2) show 2777 4218 moveto 384 ns (4) show 3028 4218 moveto 384 /Symbol f1 (\327) show 3173 4218 moveto 384 /Symbol f2 (a) show 3437 4314 moveto 224 /Times-Roman f1 (1) show 3627 4218 moveto 384 /Symbol f1 (\327) show 3772 4218 moveto 384 /Symbol f2 (a) show 4054 4314 moveto 224 /Times-Roman f1 (2) show 4262 4218 moveto 384 /Symbol f1 (\327) show 4403 4218 moveto 384 /Times-Italic f1 (sin) show 4872 4047 moveto 224 /Times-Roman f1 (2) show 5007 4218 moveto 384 /Symbol f2 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 2749 3826 moveto 2488 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 3286 1149 384 1983 4346 16 sqr 16 th 1951 3157 moveto 3350 0 rlineto stroke end pse gR gS 17 23 560 797 rC 72 175 :M f56 sf .91 .091(Namely, for each finite )J 192 167 7 10 rC 199 177 :M psb currentpoint pse 192 167 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate 64 -4583 translate -4 4839 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 199 175 :M f56 sf 1.137 .114(, the distribution of the pointer variable )J 403 158 37 30 rC 440 188 :M psb currentpoint pse 403 158 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1184 div 960 3 -1 roll exch div scale currentpoint translate -2181 -1540 translate 2609 1845 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 2823 1941 moveto 224 /Times-Roman f1 (3) show 2269 2377 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 2540 2377 moveto 384 /Symbol f1 (\327) show 2696 2377 moveto 384 /Times-Italic f1 (t) show 2829 2206 moveto 224 /Times-Roman f1 (2) show 3110 2377 moveto 384 ns (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 3103 2054 vb 16 th 2245 1985 moveto 1079 0 rlineto stroke end pse gR gS 17 23 560 797 rC 440 175 :M f56 sf 1.89 .189( in the state )J 508 165 15 15 rC 523 180 :M psb currentpoint pse 508 165 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 480 3 -1 roll exch div scale currentpoint translate -130 -1764 translate 190 2084 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 506 2180 moveto 224 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 72 201 :M f56 sf -.12(coincides with the distribution of a random variable )A 317 190 26 16 rC 343 206 :M psb currentpoint pse 317 190 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 512 3 -1 roll exch div scale currentpoint translate 64 -4520 translate -10 4872 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (x) show 212 4968 moveto 224 /Times-Roman f1 (3) show 495 4872 moveto 384 /Times-Italic f1 (t) show 352 4882 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 623 4882 moveto (\)) show end pse gR gS 17 23 560 797 rC 343 201 :M f56 sf -.128( that can be written as )A 448 189 38 18 rC 486 207 :M psb currentpoint pse 448 189 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 576 3 -1 roll exch div scale currentpoint translate 64 -4522 translate -4 4906 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 5002 moveto 224 /Times-Roman f1 (3) show 531 4906 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 5002 moveto 224 /Times-Roman f1 (0) show 388 4936 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 4936 moveto (\)) show end pse gR gS 17 23 560 797 rC 486 201 :M f56 sf -.047( plus an)A 72 222 :M -.144(independent normal random variable of mean 0 and variance:)A 188 244 215 36 rC 403 280 :M psb currentpoint pse 188 244 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6880 div 1152 3 -1 roll exch div scale currentpoint translate 64 46 translate 28 323 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (4) show 279 323 moveto 384 /Symbol f1 (\327) show 424 323 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 703 419 moveto 224 /Times-Roman f1 (3) show 825 157 moveto (2) show 1056 323 moveto 384 /Symbol f1 (+) show 1354 323 moveto 384 /Times-Italic f1 (t) show 1487 152 moveto 224 /Times-Roman f1 (2) show 81 855 moveto 384 /Symbol f2 (e) show 308 684 moveto 224 /Times-Roman f1 (2) show 516 855 moveto 384 /Symbol f1 (\327) show 661 855 moveto 384 /Symbol f2 (a) show 940 951 moveto 224 /Times-Roman f1 (3) show 1138 855 moveto 384 /Symbol f1 (\327) show 1294 855 moveto 384 /Times-Italic f1 (t) show 1431 684 moveto 224 /Times-Roman f1 (4) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 463 moveto 1657 0 rlineto stroke 1765 562 moveto 384 /Symbol f1 (=) show 2735 323 moveto 384 /Times-Roman f1 (1) show 2234 914 moveto 384 /Symbol f2 (e) show 2505 914 moveto 384 /Symbol f1 (\327) show 2661 914 moveto 384 /Times-Italic f1 (t) show 2794 743 moveto 224 /Times-Roman f1 (2) show 3075 914 moveto 384 ns (2) show 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 3068 591 vb 2095 958 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1544 /Symbol f3 (\() show 3278 958 moveto (\)) show 3410 655 moveto 224 /Times-Roman f1 (2) show 16 th 2083 463 moveto 1497 0 rlineto stroke 3896 562 moveto 384 /Symbol f1 (D) show 4131 562 moveto 384 /Times-Italic f1 (q) show 4345 658 moveto 224 /Times-Roman f1 (3) show 3751 592 moveto 384 1000 1388 /Symbol f3 (\() show 4494 592 moveto (\)) show 4626 329 moveto 224 /Times-Roman f1 (2) show 4857 562 moveto 384 /Symbol f1 (+) show 5155 562 moveto 384 /Times-Italic f1 (t) show 5288 391 moveto 224 /Times-Roman f1 (2) show 5496 562 moveto 384 /Symbol f1 (\327) show 5781 562 moveto 384 /Symbol f1 (D) show 6016 562 moveto 384 /Times-Italic f1 (p) show 6219 658 moveto 224 /Times-Roman f1 (3) show 5636 592 moveto 384 1000 1388 /Symbol f3 (\() show 6368 592 moveto (\)) show 6500 329 moveto 224 /Times-Roman f1 (2) show 3624 641 moveto 384 1000 1919 /Symbol f3 (\() show 6659 641 moveto (\)) show end pse gR gS 17 23 560 797 rC 403 263 :M f56 sf (.)S 72 311 :M -.077(Here )A 98 301 20 15 rC 118 316 :M psb currentpoint pse 98 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 480 3 -1 roll exch div scale currentpoint translate -3735 -5511 translate 3797 5831 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (D) show 4032 5831 moveto 384 /Times-Italic f1 (q) show 4246 5927 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 118 311 :M f56 sf -.081( and )A 141 301 20 15 rC 161 316 :M psb currentpoint pse 141 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 480 3 -1 roll exch div scale currentpoint translate -5620 -5511 translate 5682 5831 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (D) show 5917 5831 moveto 384 /Times-Italic f1 (p) show 6120 5927 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 161 311 :M f56 sf -.07( are, respectively, the standard deviations of )A 372 301 13 15 rC 385 316 :M psb currentpoint pse 372 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate -3977 -5511 translate 4032 5831 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 4246 5927 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 385 311 :M f56 sf -.081( and )A 408 301 13 15 rC 421 316 :M psb currentpoint pse 408 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate -5853 -5511 translate 5917 5831 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 6120 5927 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 421 311 :M f56 sf -.075( in the state )A 478 301 16 15 rC 494 316 :M psb currentpoint pse 478 301 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 480 3 -1 roll exch div scale currentpoint translate -471 -4586 translate 531 4906 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 5002 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 494 311 :M f56 sf 1.113 .111(. This)J 72 331 :M -.08(situation is depicted in Figure 3.)A 143 365 300 184 rC 300 184 8 900 [300 0 0 184 143 365] F T 1 F :f :l FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF%ADOeod :j gR gS 17 23 560 797 rC 1 65.5 352 @w 1 65.5 352 @w 452 66 352.5 @c 1 518.5 352 @w 1 518.5 352 @w 196 65.5 353 @w 1 65.5 549 @w 1 65.5 549 @w 452 66 549.5 @c 196 518.5 353 @w 1 518.5 549 @w 1 518.5 549 @w 276 559 :M f56 sf -.046(Figure 3)A 154 575 :M f193 sf -.097(The dots represent the probability mass function of )A 397 563 38 18 rC 435 581 :M psb currentpoint pse 397 563 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 576 3 -1 roll exch div scale currentpoint translate 64 -4522 translate -4 4906 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 5002 moveto 224 /Times-Roman f1 (3) show 531 4906 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 5002 moveto 224 /Times-Roman f1 (0) show 388 4936 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 4936 moveto (\)) show end pse gR gS 17 23 560 797 rC 435 575 :M f193 sf (;)S 118 593 :M -.095(the continuous graph is the probability density function of )A 394 582 13 16 rC 407 598 :M psb currentpoint pse 394 582 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 512 3 -1 roll exch div scale currentpoint translate 64 58 translate -10 294 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (x) show 212 390 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 407 593 :M f193 sf -.249( at time )A 444 583 26 12 rC 470 595 :M psb currentpoint pse 444 583 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 211 261 moveto 384 /Symbol f1 (=) show 525 261 moveto 384 /Times-Roman f1 (4) show end pse gR gS 17 23 560 797 rC 470 593 :M f193 sf (.)S 76 607 :M -.153(\(the values of the parameters are the same as in Figures 1, 2 and are summarised in Table 1\).)A 89 630 46 12 rC 135 642 :M psb currentpoint pse 89 630 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1472 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -16 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (w) show 392 261 moveto 384 /Symbol f1 (=) show 699 261 moveto 384 /Times-Roman f1 (2) show 951 261 moveto 384 /Symbol f1 (\327) show 1108 261 moveto 384 /Symbol f2 (p) show end pse gR gS 189 630 26 12 rC 215 642 :M psb currentpoint pse 189 630 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -8 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 301 261 moveto 384 /Symbol f1 (=) show 577 261 moveto 384 /Times-Roman f1 (1) show end pse gR gS 276 630 34 14 rC 310 644 :M psb currentpoint pse 276 630 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -15 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 249 357 moveto 224 /Times-Roman f1 (1) show 477 261 moveto 384 /Symbol f1 (=) show 791 261 moveto 384 /Times-Roman f1 (4) show end pse gR gS 363 630 41 14 rC 404 644 :M psb currentpoint pse 363 630 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1312 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -15 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 267 357 moveto 224 /Times-Roman f1 (2) show 513 261 moveto 384 /Symbol f1 (=) show 818 261 moveto 384 /Times-Roman f1 (64) show end pse gR gS 458 630 32 15 rC 490 645 :M psb currentpoint pse 458 630 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1024 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -15 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 264 357 moveto 224 /Times-Roman f1 (3) show 500 261 moveto 384 /Symbol f1 (=) show 776 261 moveto 384 /Times-Roman f1 (1) show end pse gR gS 17 23 560 797 rC 1 65.5 628 @w 1 65.5 628 @w 89 66 628.5 @c 1 155.5 628 @w 90 156 628.5 @c 1 246.5 628 @w 89 247 628.5 @c 1 336.5 628 @w 90 337 628.5 @c 1 427.5 628 @w 90 428 628.5 @c 1 518.5 628 @w 1 518.5 628 @w 16 65.5 629 @w 1 65.5 645 @w 1 65.5 645 @w 89 66 645.5 @c 16 155.5 629 @w 1 155.5 645 @w 90 156 645.5 @c 16 246.5 629 @w 1 246.5 645 @w 89 247 645.5 @c 16 336.5 629 @w 1 336.5 645 @w 90 337 645.5 @c 16 427.5 629 @w 1 427.5 645 @w 90 428 645.5 @c 16 518.5 629 @w 1 518.5 645 @w 1 518.5 645 @w 280 655 :M f56 sf -.219(Table 1)A 72 691 :M f67 sf .196 .02(3. The stochastic process)J 72 709 :M f56 sf -.069(The considerations of the previous Section strictly parallel Feynman's time-of-flight analysis of)A 72 727 :M -.035(linear momentum [6], that in the present notational set-up can be summarised by the following)A 72 745 :M -.16(relation between characteristic functions:)A endp showpage %%Page: 12 12 %%BeginPageSetup initializepage (defalco; pag: 12 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (12)S 72 106 :M ( )S 157 86 279 34 rC 436 120 :M psb currentpoint pse 157 86 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8928 div 1088 3 -1 roll exch div scale currentpoint translate 64 -5986 translate 142 6626 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 455 6722 moveto 224 /Times-Roman f1 (0) show 601 6626 moveto 384 /Times-Roman f1 (,) show 734 6626 moveto 384 /Times-Italic f1 (e) show 923 6412 moveto 224 ns (i) show 998 6412 moveto 224 /Symbol f1 (\327) show 1065 6412 moveto 224 /Symbol f2 (t) show 1197 6412 moveto 224 /Symbol f1 (\327) show 1266 6412 moveto 224 /Times-Italic f1 (p) show 1411 6469 moveto 160 ns (j) show 1504 6626 moveto 384 /Symbol f2 (y) show 1817 6722 moveto 224 /Times-Roman f1 (0) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -98 299 98 6228 vb 16 th 98 299 0 6527 vb 16 th 98 299 1987 6228 vb 16 th -98 299 2085 6527 vb 2209 6626 moveto 384 /Symbol f1 (=) show 2520 6626 moveto 384 /Times-Roman f1 (lim) show 2536 6850 moveto 224 /Times-Italic f1 (t) show 2620 6850 moveto 224 /Symbol f1 (\256) show 2861 6850 moveto (\245) show 3237 6626 moveto 384 /Times-Italic f1 (e) show 3428 6425 moveto 224 /Symbol f1 (-) show 3575 6425 moveto 224 /Times-Italic f1 (i) show 3650 6425 moveto 224 /Symbol f1 (\327) show 3717 6425 moveto 224 /Symbol f2 (t) show 3849 6425 moveto 224 /Symbol f1 (\327) show 3914 6425 moveto 224 /Times-Italic f1 (H) show 4100 6481 moveto 160 /Times-Roman f1 (0) show 4182 6481 moveto 160 /Times-Roman f1 (,) show 4227 6481 moveto 160 /Times-Roman f1 (0) show 4363 6626 moveto 384 /Symbol f2 (y) show 4676 6722 moveto 224 /Times-Roman f1 (0) show 4822 6626 moveto 384 /Times-Roman f1 (,) show 4955 6626 moveto (exp) show 5705 6626 moveto 384 /Times-Italic f1 (i) show 5874 6626 moveto 384 /Symbol f1 (\327) show 6031 6626 moveto 384 /Symbol f2 (t) show 6299 6626 moveto 384 /Symbol f1 (\327) show 6482 6343 moveto 384 /Times-Italic f1 (q) show 6717 6439 moveto 224 ns (j) show 6588 6919 moveto 384 ns (t) show 16 th 6459 6527 moveto 369 0 rlineto stroke 5514 6405 moveto 384 /Symbol f1 (\346) show 5514 6892 moveto (\350) show 5514 6673 moveto (\347) show 6840 6405 moveto (\366) show 6840 6892 moveto (\370) show 6840 6673 moveto (\367) show 7036 6626 moveto 384 /Times-Italic f1 (e) show 7227 6425 moveto 224 /Symbol f1 (-) show 7374 6425 moveto 224 /Times-Italic f1 (i) show 7449 6425 moveto 224 /Symbol f1 (\327) show 7516 6425 moveto 224 /Symbol f2 (t) show 7648 6425 moveto 224 /Symbol f1 (\327) show 7713 6425 moveto 224 /Times-Italic f1 (H) show 7899 6481 moveto 160 /Times-Roman f1 (0) show 7981 6481 moveto 160 /Times-Roman f1 (,) show 8026 6481 moveto 160 /Times-Roman f1 (0) show 8162 6626 moveto 384 /Symbol f2 (y) show 8475 6722 moveto 224 /Times-Roman f1 (0) show -140 509 3197 6018 vb 16 th 140 509 3057 6527 vb 16 th 140 509 8645 6018 vb 16 th -140 509 8785 6527 vb end pse gR gS 17 23 560 797 rC 437 106 :M f56 sf .045 .005(, \(11\))J 72 141 22 15 rC 94 156 :M psb currentpoint pse 72 141 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 704 div 480 3 -1 roll exch div scale currentpoint translate 64 -7209 translate -7 7529 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 289 7625 moveto 224 /Times-Roman f1 (0) show 413 7625 moveto 224 /Times-Roman f1 (,) show 485 7625 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 94 151 :M f56 sf -.109( being the free Hamiltonian.)A 72 189 :M -.045(As it is well known, the simple statement \(11\) has evolved, after the germinal work of Shucker)A 72 207 :M .649 .065([8], who translated it in terms of the pathwise asymptotics of Nelson's processes associated)J 72 225 :M -.052(with the free Schr\232dinger evolution, into a powerful stochastic model of quantum scattering by)A 72 243 :M 1.119 .112(a scalar potential [9,10]. In the same spirit, by studying, in this Section, Nelson's process)J 72 260 :M 1.246 .125(associated with )J 153 247 78 18 rC 231 265 :M psb currentpoint pse 153 247 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2496 div 576 3 -1 roll exch div scale currentpoint translate 64 -4629 translate -4 5045 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 5141 moveto 224 /Times-Italic f1 (t) show 514 5045 moveto 384 /Symbol f1 (=) show 824 5045 moveto 384 /Times-Italic f1 (e) show 1015 4844 moveto 224 /Symbol f1 (-) show 1162 4844 moveto 224 /Times-Italic f1 (i) show 1237 4844 moveto 224 /Symbol f1 (\327) show 1304 4844 moveto 224 /Times-Italic f1 (t) show 1383 4844 moveto 224 /Symbol f1 (\327) show 1448 4844 moveto 224 /Times-Italic f1 (H) show 1631 4900 moveto 160 /Symbol f2 (w) show 1762 4900 moveto 160 /Times-Roman f1 (,) show 1807 4900 moveto 160 /Symbol f2 (e) show 1951 5045 moveto 384 ns (y) show 2264 5141 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 232 260 :M f56 sf 1.807 .181( \(for the simple choice \(9\) of )J 391 250 16 15 rC 407 265 :M psb currentpoint pse 391 250 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 480 3 -1 roll exch div scale currentpoint translate -82 -6306 translate 142 6626 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 455 6722 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 407 260 :M f56 sf 1.597 .16( dictated by criteria of)J 72 280 :M 1.809 .181(feasibility of numerical simulations\) we intend to contribute to a preliminary, heuristic)J 72 298 :M -.096(understanding of some aspects of stochastic motion in a vector potential.)A 72 334 :M .63 .063(For an analysis of unsurpassed clarity of the Hamiltonian approach to stochastic mechanics)J 72 352 :M -.065(needed in the case, considered here, of velocity dependent forces, we refer the reader to [11].)A 72 370 :M -.036(Here we just recall the basic facts we need in order to draw Figure 1, namely in order to define)A 72 388 :M -.116(and simulate the stochastic process)A 238 410 119 18 rC 357 428 :M psb currentpoint pse 238 410 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3808 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate -9 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 328 moveto (t) show 194 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 338 moveto (\)) show 679 328 moveto 384 /Symbol f1 (=) show 1115 328 moveto 384 /Times-Italic f1 (q) show 1314 424 moveto 224 /Times-Roman f1 (1) show 1589 328 moveto 384 /Times-Italic f1 (t) show 1446 338 moveto 384 1000 1171 /Symbol f3 (\() show 1717 338 moveto (\)) show 1831 328 moveto 384 /Times-Roman f1 (,) show 1963 328 moveto 384 /Times-Italic f1 (q) show 2180 424 moveto 224 /Times-Roman f1 (2) show 2473 328 moveto 384 /Times-Italic f1 (t) show 2330 338 moveto 384 1000 1171 /Symbol f3 (\() show 2601 338 moveto (\)) show 2715 328 moveto 384 /Times-Roman f1 (,) show 2847 328 moveto 384 /Times-Italic f1 (q) show 3061 424 moveto 224 /Times-Roman f1 (3) show 3344 328 moveto 384 /Times-Italic f1 (t) show 3201 338 moveto 384 1000 1171 /Symbol f3 (\() show 3472 338 moveto (\)) show 977 358 moveto 384 1000 1388 /Symbol f3 (\() show 3599 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 460 :M f56 sf -.061(corresponding to )A 155 447 78 18 rC 233 465 :M psb currentpoint pse 155 447 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2496 div 576 3 -1 roll exch div scale currentpoint translate 64 -4629 translate -4 5045 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 5141 moveto 224 /Times-Italic f1 (t) show 514 5045 moveto 384 /Symbol f1 (=) show 824 5045 moveto 384 /Times-Italic f1 (e) show 1015 4844 moveto 224 /Symbol f1 (-) show 1162 4844 moveto 224 /Times-Italic f1 (i) show 1237 4844 moveto 224 /Symbol f1 (\327) show 1304 4844 moveto 224 /Times-Italic f1 (t) show 1383 4844 moveto 224 /Symbol f1 (\327) show 1448 4844 moveto 224 /Times-Italic f1 (H) show 1631 4900 moveto 160 /Symbol f2 (w) show 1762 4900 moveto 160 /Times-Roman f1 (,) show 1807 4900 moveto 160 /Symbol f2 (e) show 1951 5045 moveto 384 ns (y) show 2264 5141 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 234 460 :M f56 sf .257 .026(, with )J 265 450 24 15 rC 289 465 :M psb currentpoint pse 265 450 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 768 div 480 3 -1 roll exch div scale currentpoint translate 64 -6751 translate -7 7071 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 285 7167 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (w) show 478 7167 moveto 224 /Times-Roman f1 (,) show 551 7167 moveto 224 /Symbol f2 (e) show end pse gR gS 17 23 560 797 rC 289 460 :M f56 sf -.057( given by \(1\) and )A 373 450 16 15 rC 389 465 :M psb currentpoint pse 373 450 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 512 div 480 3 -1 roll exch div scale currentpoint translate -471 -4586 translate 531 4906 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 5002 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 389 460 :M f56 sf .082 .008( given by \(4\).)J 72 497 :M -.114(The probability density of )A 197 486 24 15 rC 221 501 :M psb currentpoint pse 197 486 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 768 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 332 292 moveto 384 /Times-Roman f1 (0) show 194 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 535 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 221 497 :M f56 sf -.084( is :)A 256 502 83 20 rC 339 522 :M psb currentpoint pse 256 502 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2656 div 640 3 -1 roll exch div scale currentpoint translate 64 62 translate -19 386 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (r) show 356 386 moveto 384 /Times-Roman f1 (0) show 546 386 moveto 384 /Times-Roman f1 (,) show 681 386 moveto 384 /Times-Italic f1 (x) show 218 396 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 885 396 moveto (\)) show 1099 386 moveto 384 /Symbol f1 (=) show 1461 386 moveto 384 /Symbol f2 (y) show 1774 482 moveto 224 /Times-Roman f1 (0) show 2065 386 moveto 384 /Times-Italic f1 (x) show 1924 396 moveto 384 1000 1171 /Symbol f3 (\() show 2269 396 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1425 57 moveto 0 460 rlineto stroke 2415 57 moveto 0 460 rlineto stroke 2449 152 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 72 553 :M f56 sf -.016(where, of course, we use the notation )A 253 541 70 18 rC 323 559 :M psb currentpoint pse 253 541 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2240 div 576 3 -1 roll exch div scale currentpoint translate 64 -693 translate -6 1077 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 285 1077 moveto 384 /Symbol f1 (=) show 724 1077 moveto 384 /Times-Italic f1 (x) show 915 1173 moveto 224 /Times-Roman f1 (1) show 1043 1077 moveto 384 /Times-Roman f1 (,) show 1178 1077 moveto 384 /Times-Italic f1 (x) show 1387 1173 moveto 224 /Times-Roman f1 (2) show 1533 1077 moveto 384 /Times-Roman f1 (,) show 1668 1077 moveto 384 /Times-Italic f1 (x) show 1874 1173 moveto 224 /Times-Roman f1 (3) show 583 1107 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 2023 1107 moveto (\)) show end pse gR gS 17 23 560 797 rC 323 553 :M f56 sf (.)S 72 591 :M -.097(The components of the mean forward velocity field are given, for )A 383 581 43 14 rC 426 595 :M psb currentpoint pse 383 581 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 -1 roll exch div scale currentpoint translate 64 -1381 translate 20 1701 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (j) show 209 1701 moveto 384 /Symbol f1 (=) show 485 1701 moveto 384 /Times-Roman f1 (1) show 644 1701 moveto 384 /Times-Roman f1 (,) show 774 1701 moveto 384 /Times-Roman f1 (2) show 964 1701 moveto 384 /Times-Roman f1 (,) show 1089 1701 moveto 384 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 426 591 :M f56 sf .266 .027( , by:)J 150 614 295 38 rC 445 652 :M psb currentpoint pse 150 614 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 9440 div 1216 3 -1 roll exch div scale currentpoint translate 64 -1979 translate -3 2683 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 223 2779 moveto 224 ns (j) show 457 2683 moveto 384 ns (t) show 572 2683 moveto 384 /Times-Roman f1 (,) show 707 2683 moveto 384 /Times-Italic f1 (x) show 314 2693 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 911 2693 moveto (\)) show 1125 2683 moveto 384 /Symbol f1 (=) show 1439 2683 moveto 384 /Times-Roman f1 (Re) show 2442 2444 moveto 384 /Times-Roman f1 (1) show 2100 2976 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 2416 3072 moveto 224 /Times-Italic f1 (t) show 2663 2976 moveto 384 ns (x) show 2522 2986 moveto 384 1000 1171 /Symbol f3 (\() show 2867 2986 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2072 2584 moveto 933 0 rlineto stroke 3075 2683 moveto 384 /Symbol f1 (\327) show 3258 2444 moveto 384 /Symbol f2 (\266) show 3478 2444 moveto (y) show 3794 2540 moveto 224 /Times-Italic f1 (t) show 4041 2444 moveto 384 ns (x) show 3900 2454 moveto 384 1000 1171 /Symbol f3 (\() show 4245 2454 moveto (\)) show 3542 2976 moveto 384 /Symbol f2 (\266) show 3762 2976 moveto 384 /Times-Italic f1 (x) show 3989 3072 moveto 224 ns (j) show 3235 2584 moveto 1148 0 rlineto stroke 1873 2371 moveto 384 /Symbol f1 (\346) show 1873 3040 moveto (\350) show 1873 2821 moveto (\347) show 4395 2371 moveto (\366) show 4395 3040 moveto (\370) show 4395 2821 moveto (\367) show 4660 2683 moveto (+) show 4956 2683 moveto 384 /Times-Roman f1 (Im) show 5962 2444 moveto 384 /Times-Roman f1 (1) show 5620 2976 moveto 384 /Symbol f2 (y) show 5936 3072 moveto 224 /Times-Italic f1 (t) show 6183 2976 moveto 384 ns (x) show 6042 2986 moveto 384 1000 1171 /Symbol f3 (\() show 6387 2986 moveto (\)) show 5592 2584 moveto 933 0 rlineto stroke 6595 2683 moveto 384 /Symbol f1 (\327) show 6778 2444 moveto 384 /Symbol f2 (\266) show 6998 2444 moveto (y) show 7314 2540 moveto 224 /Times-Italic f1 (t) show 7561 2444 moveto 384 ns (x) show 7420 2454 moveto 384 1000 1171 /Symbol f3 (\() show 7765 2454 moveto (\)) show 7062 2976 moveto 384 /Symbol f2 (\266) show 7282 2976 moveto 384 /Times-Italic f1 (x) show 7509 3072 moveto 224 ns (j) show 6755 2584 moveto 1148 0 rlineto stroke 5393 2371 moveto 384 /Symbol f1 (\346) show 5393 3040 moveto (\350) show 5393 2821 moveto (\347) show 7915 2371 moveto (\366) show 7915 3040 moveto (\370) show 7915 2821 moveto (\367) show 8179 2683 moveto (-) show 8475 2683 moveto 384 /Times-Italic f1 (A) show 8790 2779 moveto 224 ns (j) show 9022 2683 moveto 384 ns (x) show 8881 2693 moveto 384 1000 1171 /Symbol f3 (\() show 9226 2693 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 684 :M f56 sf -.181(The vector potential )A 168 673 26 15 rC 194 688 :M psb currentpoint pse 168 673 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -5 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (A) show 419 292 moveto (x) show 278 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 623 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 194 684 :M f56 sf -.083( is given by:)A 247 737 :M ( )S 250 707 98 54 rC 348 761 :M psb currentpoint pse 250 707 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3136 div 1728 3 -1 roll exch div scale currentpoint translate 64 -6023 translate 190 6407 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (A) show 469 6503 moveto 224 /Times-Roman f1 (1) show 742 6407 moveto 384 /Times-Italic f1 (x) show 601 6417 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 946 6417 moveto (\)) show 1160 6407 moveto 384 /Symbol f1 (=) show 1474 6407 moveto (-) show 1703 6407 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 1974 6407 moveto 384 /Symbol f1 (\327) show 2128 6407 moveto 384 /Times-Italic f1 (x) show 2337 6503 moveto 224 /Times-Roman f1 (2) show 2545 6407 moveto 384 /Symbol f1 (\327) show 2699 6407 moveto 384 /Times-Italic f1 (x) show 2905 6503 moveto 224 /Times-Roman f1 (3) show 259 6983 moveto 384 /Times-Italic f1 (A) show 556 7079 moveto 224 /Times-Roman f1 (2) show 847 6983 moveto 384 /Times-Italic f1 (x) show 706 6993 moveto 384 1000 1171 /Symbol f3 (\() show 1051 6993 moveto (\)) show 1265 6983 moveto 384 /Symbol f1 (=) show 1575 6983 moveto 384 /Symbol f2 (e) show 1846 6983 moveto 384 /Symbol f1 (\327) show 2000 6983 moveto 384 /Times-Italic f1 (x) show 2191 7079 moveto 224 /Times-Roman f1 (1) show 2381 6983 moveto 384 /Symbol f1 (\327) show 2535 6983 moveto 384 /Times-Italic f1 (x) show 2741 7079 moveto 224 /Times-Roman f1 (3) show 2869 6983 moveto 384 /Times-Roman f1 ( ) show 278 7559 moveto 384 /Times-Italic f1 (A) show 572 7655 moveto 224 /Times-Roman f1 (3) show 853 7559 moveto 384 /Times-Italic f1 (x) show 712 7569 moveto 384 1000 1171 /Symbol f3 (\() show 1057 7569 moveto (\)) show 1271 7559 moveto 384 /Symbol f1 (=) show 1580 7559 moveto 384 /Times-Roman f1 (0) show 1794 7559 moveto 384 /Times-Roman f1 ( ) show -5 6433 moveto 384 /Symbol f1 (\354) show -5 7048 moveto (\355) show -5 6756 moveto (\357) show -5 7664 moveto (\356) show -5 7422 moveto (\357) show end pse endp showpage %%Page: 13 13 %%BeginPageSetup initializepage (defalco; pag: 13 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (13)S 72 111 :M -.081(The system of stochastic differential equations is, for )A 328 101 43 14 rC 371 115 :M psb currentpoint pse 328 101 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 -1 roll exch div scale currentpoint translate 64 -1381 translate 20 1701 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (j) show 209 1701 moveto 384 /Symbol f1 (=) show 485 1701 moveto 384 /Times-Roman f1 (1) show 644 1701 moveto 384 /Times-Roman f1 (,) show 774 1701 moveto 384 /Times-Roman f1 (2) show 964 1701 moveto 384 /Times-Roman f1 (,) show 1089 1701 moveto 384 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 371 111 :M f56 sf (:)S 72 146 :M ( )S 228 134 147 18 rC 375 152 :M psb currentpoint pse 228 134 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4704 div 576 3 -1 roll exch div scale currentpoint translate 64 -7862 translate -9 8246 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (dq) show 418 8342 moveto 224 ns (j) show 652 8246 moveto 384 ns (t) show 509 8256 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 780 8256 moveto (\)) show 994 8246 moveto 384 /Symbol f1 (=) show 1309 8246 moveto 384 /Times-Italic f1 (b) show 1535 8342 moveto 224 ns (j) show 1769 8246 moveto 384 ns (t) show 1884 8246 moveto 384 /Times-Roman f1 (,) show 2016 8246 moveto 384 /Times-Italic f1 (q) show 2362 8246 moveto (t) show 2219 8256 moveto 384 1000 1171 /Symbol f3 (\() show 2490 8256 moveto (\)) show 1626 8274 moveto 384 1000 1364 /Symbol f3 (\() show 2617 8274 moveto (\)) show 2793 8246 moveto 384 /Symbol f1 (\327) show 2944 8246 moveto 384 /Times-Italic f1 (dt) show 3336 8246 moveto 384 /Symbol f1 (+) show 3629 8246 moveto 384 /Times-Italic f1 (dw) show 4131 8342 moveto 224 ns (j) show 4365 8246 moveto 384 ns (t) show 4222 8256 moveto 384 1000 1171 /Symbol f3 (\() show 4493 8256 moveto (\)) show end pse gR gS 17 23 560 797 rC 375 146 :M f56 sf ( \(12\))S 72 182 :M -.077(Here )A 98 172 31 12 rC 129 184 :M psb currentpoint pse 98 172 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 992 div 384 3 -1 roll exch div scale currentpoint translate 64 -8563 translate -9 8883 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (dt) show 393 8883 moveto 384 /Symbol f1 (>) show 696 8883 moveto 384 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 129 182 :M f56 sf -.069(, and the independent Brownian motions )A 324 171 28 17 rC 352 188 :M psb currentpoint pse 324 171 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 896 div 544 3 -1 roll exch div scale currentpoint translate -3765 -7894 translate 3821 8246 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (w) show 4131 8342 moveto 224 ns (j) show 4365 8246 moveto 384 ns (t) show 4222 8256 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 4493 8256 moveto (\)) show end pse gR gS 17 23 560 797 rC 352 182 :M f56 sf -.126( are such that)A 186 207 223 20 rC 409 227 :M psb currentpoint pse 186 207 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7136 div 640 3 -1 roll exch div scale currentpoint translate 64 -8579 translate -4 8995 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (E) show 375 8995 moveto (dw) show 877 9091 moveto 224 ns (j) show 1111 8995 moveto 384 ns (t) show 968 9005 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1239 9005 moveto (\)) show 237 9050 moveto 384 1000 1661 /Symbol f3 (\() show 1366 9050 moveto (\)) show 1580 8995 moveto 384 /Symbol f1 (=) show 1889 8995 moveto 384 /Times-Roman f1 (0) show 2103 8995 moveto 384 /Times-Roman f1 ( and ) show 2849 8995 moveto 384 /Times-Italic f1 (E) show 3228 8995 moveto (dw) show 3730 9091 moveto 224 ns (j) show 3964 8995 moveto 384 ns (t) show 3821 9005 moveto 384 1000 1171 /Symbol f3 (\() show 4092 9005 moveto (\)) show 4268 8995 moveto 384 /Symbol f1 (\327) show 4419 8995 moveto 384 /Times-Italic f1 (dw) show 4909 9091 moveto 224 ns (k) show 5208 8995 moveto 384 ns (t) show 5065 9005 moveto 384 1000 1171 /Symbol f3 (\() show 5336 9005 moveto (\)) show 3090 9050 moveto 384 1000 1661 /Symbol f3 (\() show 5463 9050 moveto (\)) show 5670 8995 moveto 384 /Times-Roman f1 (=) show 5972 8995 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (d) show 6220 9091 moveto 224 /Times-Roman f1 (j) show 6285 9091 moveto (,) show 6362 9091 moveto (k) show 6576 8995 moveto 384 /Symbol f1 (\327) show 6727 8995 moveto 384 /Times-Italic f1 (dt) show end pse gR gS 17 23 560 797 rC 72 260 :M f56 sf -.072(The construction leading to the numerical simulation of \(12\) and, therefore, to the sample paths)A 72 278 :M -.002(of Figure 1 is completed by the following explicit expression for the solution, under the initial)A 72 294 :M -.07(condition \(9\), of the Schr\232dinger equation )A 274 284 94 15 rC 368 299 :M psb currentpoint pse 274 284 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3008 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -8 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (i) show 161 261 moveto 384 /Symbol f1 (\327) show 312 261 moveto 384 /Times-Italic f1 (d) show 504 261 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 820 357 moveto 224 /Times-Italic f1 (t) show 1096 261 moveto 384 ns (dt) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -115 359 1059 0 vb 1503 261 moveto 384 /Symbol f1 (=) show 1814 261 moveto 384 /Times-Italic f1 (H) show 2106 357 moveto 224 /Symbol f2 (w) show 2299 357 moveto 224 /Times-Roman f1 (,) show 2372 357 moveto 224 /Symbol f2 (e) show 2517 261 moveto 384 ns (y) show 2833 357 moveto 224 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 368 294 :M f56 sf -.334( :)A 186 318 218 32 rC 404 350 :M psb currentpoint pse 186 318 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6976 div 1024 3 -1 roll exch div scale currentpoint translate 64 54 translate -4 554 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 650 moveto 224 /Times-Italic f1 (t) show 559 554 moveto 384 ns (x) show 418 564 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 763 564 moveto (\)) show 977 554 moveto 384 /Symbol f1 (=) show 1839 554 moveto 384 /Times-Italic f1 (c) show 2152 554 moveto (t) show 2267 554 moveto 384 /Times-Roman f1 (,) show 2402 554 moveto 384 /Times-Italic f1 (x) show 2608 650 moveto 224 /Times-Roman f1 (3) show 2744 554 moveto 384 /Times-Roman f1 (;) show 2874 554 moveto 384 /Times-Roman f1 (2) show 3126 554 moveto 384 /Symbol f1 (\327) show 3278 554 moveto 384 /Symbol f2 (e) show 3549 554 moveto 384 /Symbol f1 (\327) show 3706 554 moveto 384 /Times-Italic f1 (n) show 3902 554 moveto 384 /Times-Roman f1 (,) show 4290 554 moveto 384 /Symbol f2 (a) show 4569 650 moveto 224 /Times-Roman f1 (3) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 686 472 384 4043 685 16 sqr 2009 608 moveto 384 1000 1653 /Symbol f3 (\() show 4750 608 moveto (\)) show 4926 554 moveto 384 /Symbol f1 (\327) show 5079 554 moveto 384 /Times-Italic f1 (K) show 5381 650 moveto 224 ns (n) show 5677 554 moveto 384 ns (t) show 5792 554 moveto 384 /Times-Roman f1 (,) show 5927 554 moveto 384 /Times-Italic f1 (x) show 6118 650 moveto 224 /Times-Roman f1 (1) show 6246 554 moveto 384 /Times-Roman f1 (,) show 6381 554 moveto 384 /Times-Italic f1 (x) show 6590 650 moveto 224 /Times-Roman f1 (2) show 5534 582 moveto 384 1000 1369 /Symbol f3 (\() show 6749 582 moveto (\)) show 1293 927 moveto 224 /Times-Italic f1 (n) show 1432 927 moveto 224 /Symbol f1 (=) show 1582 927 moveto (-) show 1728 927 moveto (\245) show 1437 152 moveto (+) show 1584 152 moveto (\245) show 1384 641 moveto 576 ns (\345) show end pse gR gS 17 23 560 797 rC 404 337 :M f56 sf (.)S 72 383 :M -.103(Here)A 99 405 396 43 rC 495 448 :M psb currentpoint pse 99 405 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12672 div 1376 3 -1 roll exch div scale currentpoint translate 64 -1097 translate -9 1897 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (c) show 304 1897 moveto (t) show 419 1897 moveto 384 /Times-Roman f1 (,) show 551 1897 moveto 384 /Times-Italic f1 (z) show 719 1897 moveto 384 /Times-Roman f1 (;) show 853 1897 moveto 384 /Times-Italic f1 (f) show 1008 1897 moveto 384 /Times-Roman f1 (,) show 1138 1897 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (s) show 161 1907 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1422 1907 moveto (\)) show 1636 1897 moveto 384 /Symbol f1 (=) show 3730 1658 moveto 384 /Times-Roman f1 (1) show 2102 2210 moveto (2) show 2354 2210 moveto 384 /Symbol f1 (\327) show 2511 2210 moveto 384 /Symbol f2 (p) show 1966 2220 moveto 384 1000 1171 /Symbol f3 (\() show 2772 2220 moveto (\)) show 2886 2013 moveto 224 /Times-Roman f1 (1) show 3063 2013 moveto (4) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -64 200 3040 1861 vb 3483 2210 moveto 384 /Symbol f2 (s) show 3839 2210 moveto 384 /Symbol f1 (+) show 4133 2210 moveto 384 /Times-Italic f1 (i) show 4302 2210 moveto 384 /Symbol f1 (\327) show 4458 2210 moveto 384 /Times-Italic f1 (t) show 4844 2210 moveto 384 /Times-Roman f1 (2) show 5096 2210 moveto 384 /Symbol f1 (\327) show 5245 2210 moveto 384 /Symbol f2 (s) show 4708 2220 moveto 384 1000 1171 /Symbol f3 (\() show 5529 2220 moveto (\)) show 8 th -124 387 4690 1918 vb /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 2435 499 384 3232 2337 16 sqr 16 th 1954 1798 moveto 3745 0 rlineto stroke 5769 1897 moveto 384 /Symbol f1 (\327) show 5921 1897 moveto 384 /Times-Roman f1 (exp) show 6675 1897 moveto 384 /Symbol f1 (-) show 7154 1602 moveto 384 /Times-Italic f1 (z) show 7406 1602 moveto 384 /Symbol f1 (-) show 7700 1602 moveto 384 /Times-Italic f1 (f) show 7917 1602 moveto 384 /Symbol f1 (\327) show 8073 1602 moveto 384 /Times-Italic f1 (t) show 8206 1431 moveto 224 /Times-Roman f1 (2) show 8486 1602 moveto 384 ns (2) show 8 th -130 406 8479 1279 vb 7016 1646 moveto 384 1000 1544 /Symbol f3 (\() show 8689 1646 moveto (\)) show 8821 1343 moveto 224 /Times-Roman f1 (2) show 6972 2190 moveto 384 ns (4) show 7223 2190 moveto 384 /Symbol f1 (\327) show 7372 2190 moveto 384 /Symbol f2 (s) show 7661 2019 moveto 224 /Times-Roman f1 (2) show 7892 2190 moveto 384 /Symbol f1 (+) show 8183 2190 moveto 384 /Times-Roman f1 (2) show 8435 2190 moveto 384 /Symbol f1 (\327) show 8587 2190 moveto 384 /Times-Italic f1 (i) show 8756 2190 moveto 384 /Symbol f1 (\327) show 8912 2190 moveto 384 /Times-Italic f1 (t) show 16 th 6944 1798 moveto 2107 0 rlineto stroke 9144 1897 moveto 384 /Symbol f1 (+) show 9438 1897 moveto 384 /Times-Italic f1 (i) show 9607 1897 moveto 384 /Symbol f1 (\327) show 9885 1897 moveto 384 /Times-Italic f1 (z) show 10115 1897 moveto 384 /Symbol f1 (\327) show 10268 1897 moveto 384 /Times-Italic f1 (f) show 10485 1897 moveto 384 /Symbol f1 (\327) show 10641 1897 moveto 384 /Times-Italic f1 (t) show 10840 1897 moveto 384 /Symbol f1 (-) show 11134 1897 moveto 384 /Times-Italic f1 (f) show 11307 1726 moveto 224 /Times-Roman f1 (2) show 11515 1897 moveto 384 /Symbol f1 (\327) show 11671 1897 moveto 384 /Times-Italic f1 (t) show 11801 1726 moveto 224 /Times-Roman f1 (3) show 12077 1897 moveto 384 ns (6) show -130 406 12064 1574 vb 9747 1941 moveto 384 1000 1544 /Symbol f3 (\() show 12277 1941 moveto (\)) show 6480 1514 moveto 384 /Symbol f1 (\346) show 6480 2325 moveto (\350) show 6480 1965 moveto (\347) show 12395 1514 moveto (\366) show 12395 2325 moveto (\370) show 12395 1965 moveto (\367) show end pse gR gS 17 23 560 797 rC 72 481 :M f56 sf -.163(and)A 139 503 316 67 rC 455 570 :M psb currentpoint pse 139 503 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10112 div 2144 3 -1 roll exch div scale currentpoint translate 64 -1675 translate -7 2117 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (K) show 295 2213 moveto 224 ns (n) show 591 2117 moveto 384 ns (t) show 706 2117 moveto 384 /Times-Roman f1 (,) show 841 2117 moveto 384 /Times-Italic f1 (x) show 1032 2213 moveto 224 /Times-Roman f1 (1) show 1160 2117 moveto 384 /Times-Roman f1 (,) show 1295 2117 moveto 384 /Times-Italic f1 (x) show 1504 2213 moveto 224 /Times-Roman f1 (2) show 448 2145 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1369 /Symbol f3 (\() show 1663 2145 moveto (\)) show 1877 2117 moveto 384 /Symbol f1 (=) show 2187 2117 moveto 384 /Times-Italic f1 (i) show 2316 1946 moveto 224 ns (n) show 2527 2117 moveto 384 /Symbol f1 (\327) show 2673 2117 moveto 384 /Times-Italic f1 (a) show 2850 2213 moveto 224 /Times-Roman f1 (1) show 3125 2117 moveto 384 /Times-Italic f1 (t) show 2982 2127 moveto 384 1000 1171 /Symbol f3 (\() show 3253 2127 moveto (\)) show 3429 2117 moveto 384 /Symbol f1 (\327) show 3575 2117 moveto 384 /Times-Italic f1 (a) show 3770 2213 moveto 224 /Times-Roman f1 (2) show 4063 2117 moveto 384 /Times-Italic f1 (t) show 3920 2127 moveto 384 1000 1171 /Symbol f3 (\() show 4191 2127 moveto (\)) show 4367 2117 moveto 384 /Symbol f1 (\327) show 4519 2117 moveto 384 /Times-Roman f1 (exp) show 5216 2117 moveto 384 /Symbol f1 (-) show 5603 2117 moveto 384 /Times-Italic f1 (d) show 5799 2213 moveto 224 /Times-Roman f1 (1) show 6074 2117 moveto 384 /Times-Italic f1 (t) show 5931 2127 moveto 384 1000 1171 /Symbol f3 (\() show 6202 2127 moveto (\)) show 6401 2117 moveto 384 /Symbol f1 (+) show 6694 2117 moveto 384 /Times-Italic f1 (d) show 6908 2213 moveto 224 /Times-Roman f1 (2) show 7201 2117 moveto 384 /Times-Italic f1 (t) show 7058 2127 moveto 384 1000 1171 /Symbol f3 (\() show 7329 2127 moveto (\)) show 5465 2145 moveto 384 1000 1369 /Symbol f3 (\() show 7456 2145 moveto (\)) show 7632 2117 moveto 384 /Symbol f1 (\327) show 7913 2117 moveto 384 /Times-Italic f1 (x) show 8104 2213 moveto 224 /Times-Roman f1 (1) show 8218 1951 moveto (2) show 8449 2117 moveto 384 /Symbol f1 (+) show 8745 2117 moveto 384 /Times-Italic f1 (x) show 8954 2213 moveto 224 /Times-Roman f1 (2) show 9086 1951 moveto (2) show 7772 2159 moveto 384 1000 1515 /Symbol f3 (\() show 9245 2159 moveto (\)) show 9555 2117 moveto 384 /Times-Roman f1 (4) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -161 501 9542 1768 vb 5073 2176 moveto 384 1000 1708 /Symbol f3 (\() show 9757 2176 moveto (\)) show 9933 2117 moveto 384 /Symbol f1 (\327) show 0 3286 moveto 384 /Times-Roman f1 ( ) show 1990 3286 moveto 384 /Symbol f1 (\327) show 2147 3286 moveto 384 /Times-Italic f1 (J) show 2321 3382 moveto 224 ns (n) show 2613 3286 moveto 384 ns (i) show 2782 3286 moveto 384 /Symbol f1 (\327) show 3060 3286 moveto 384 /Times-Italic f1 (d) show 3256 3382 moveto 224 /Times-Roman f1 (1) show 3531 3286 moveto 384 /Times-Italic f1 (t) show 3388 3296 moveto 384 1000 1171 /Symbol f3 (\() show 3659 3296 moveto (\)) show 3857 3286 moveto 384 /Symbol f1 (-) show 4149 3286 moveto 384 /Times-Italic f1 (d) show 4363 3382 moveto 224 /Times-Roman f1 (2) show 4656 3286 moveto 384 /Times-Italic f1 (t) show 4513 3296 moveto 384 1000 1171 /Symbol f3 (\() show 4784 3296 moveto (\)) show 2922 3314 moveto 384 1000 1369 /Symbol f3 (\() show 4911 3314 moveto (\)) show 5087 3286 moveto 384 /Symbol f1 (\327) show 5368 3286 moveto 384 /Times-Italic f1 (x) show 5559 3382 moveto 224 /Times-Roman f1 (1) show 5673 3120 moveto (2) show 5904 3286 moveto 384 /Symbol f1 (+) show 6200 3286 moveto 384 /Times-Italic f1 (x) show 6409 3382 moveto 224 /Times-Roman f1 (2) show 6541 3120 moveto (2) show 5227 3328 moveto 384 1000 1515 /Symbol f3 (\() show 6700 3328 moveto (\)) show 7010 3286 moveto 384 /Times-Roman f1 (4) show 16 th -161 501 6997 2937 vb 2474 3345 moveto 384 1000 1708 /Symbol f3 (\() show 7212 3345 moveto (\)) show 7388 3286 moveto 384 /Symbol f1 (\327) show 7876 3047 moveto 384 /Times-Italic f1 (x) show 8067 3143 moveto 224 /Times-Roman f1 (1) show 8280 3047 moveto 384 /Symbol f1 (+) show 8574 3047 moveto 384 /Times-Italic f1 (i) show 8743 3047 moveto 384 /Symbol f1 (\327) show 8897 3047 moveto 384 /Times-Italic f1 (x) show 9106 3143 moveto 224 /Times-Roman f1 (2) show 8020 3625 moveto 384 /Times-Italic f1 (x) show 8211 3721 moveto 224 /Times-Roman f1 (1) show 8325 3459 moveto (2) show 8556 3625 moveto 384 /Symbol f1 (+) show 8852 3625 moveto 384 /Times-Italic f1 (x) show 9061 3721 moveto 224 /Times-Roman f1 (2) show 9193 3459 moveto (2) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1599 526 384 7764 3753 16 sqr 16 th 7732 3187 moveto 1663 0 rlineto stroke 7533 2944 moveto 384 /Symbol f1 (\346) show 7533 3673 moveto (\350) show 7533 3395 moveto (\347) show 9407 2944 moveto (\366) show 9407 3673 moveto (\370) show 9407 3395 moveto (\367) show 9605 2684 moveto 224 /Times-Roman f1 (2) show 9728 2684 moveto 224 /Symbol f1 (\327) show 9795 2684 moveto 224 /Times-Italic f1 (n) show end pse gR gS 17 23 560 797 rC 72 602 :M f56 sf -.077(Here )A 98 592 12 14 rC 110 606 :M psb currentpoint pse 98 592 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 384 div 448 3 -1 roll exch div scale currentpoint translate -9220 -12516 translate 9281 12836 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (J) show 9455 12932 moveto 224 ns (n) show end pse gR gS 17 23 560 797 rC 110 602 :M f56 sf -.047( is a Bessel function of the first kind and, for )A 325 592 33 14 rC 358 606 :M psb currentpoint pse 325 592 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1056 div 448 3 -1 roll exch div scale currentpoint translate 64 -6822 translate 20 7142 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (j) show 209 7142 moveto 384 /Symbol f1 (=) show 485 7142 moveto 384 /Times-Roman f1 (1) show 644 7142 moveto 384 /Times-Roman f1 (,) show 774 7142 moveto 384 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 358 602 :M f56 sf .455 .045( , )J 368 589 68 20 rC 436 609 :M psb currentpoint pse 368 589 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2176 div 640 3 -1 roll exch div scale currentpoint translate 64 -6262 translate -23 6678 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (b) show 238 6774 moveto 224 /Times-Italic f1 (j) show 425 6678 moveto 384 /Symbol f1 (=) show 701 6678 moveto 384 /Times-Roman f1 (1) show 1164 6678 moveto (2) show 1416 6678 moveto 384 /Symbol f1 (\327) show 1561 6678 moveto 384 /Symbol f2 (a) show 1861 6774 moveto 224 /Times-Italic f1 (j) show 1028 6733 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1661 /Symbol f3 (\() show 1961 6733 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -176 549 1002 6305 vb end pse gR gS 17 23 560 797 rC 436 602 :M f56 sf .909 .091( ,)J 171 628 252 40 rC 423 668 :M psb currentpoint pse 171 628 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8064 div 1280 3 -1 roll exch div scale currentpoint translate 64 -8772 translate -14 9540 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (a) show 199 9636 moveto 224 ns (j) show 433 9540 moveto 384 ns (t) show 290 9550 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 561 9550 moveto (\)) show 775 9540 moveto 384 /Symbol f1 (=) show 1197 9540 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (b) show 1458 9636 moveto 224 /Times-Italic f1 (j) show 1740 9540 moveto 384 /Symbol f2 (p) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -138 428 1705 9256 vb 1073 9595 moveto 384 1000 1661 /Symbol f3 (\() show 2001 9595 moveto (\)) show 2115 9262 moveto 224 /Times-Roman f1 (1) show 2237 9262 moveto 224 /Times-Roman f1 (/) show 2334 9262 moveto 224 /Times-Roman f1 (4) show 2541 9540 moveto 384 /Symbol f1 (\327) show 5005 9301 moveto 384 /Symbol f2 (w) show 2901 9833 moveto (w) show 3271 9833 moveto 384 /Symbol f1 (\327) show 3422 9833 moveto 384 /Times-Roman f1 (cos) show 4060 9833 moveto 384 /Symbol f2 (w) show 4430 9833 moveto 384 /Symbol f1 (\327) show 4586 9833 moveto 384 /Times-Italic f1 (t) show 3929 9843 moveto 384 1000 1171 /Symbol f3 (\() show 4714 9843 moveto (\)) show 4913 9833 moveto 384 /Symbol f1 (+) show 5207 9833 moveto 384 /Times-Italic f1 (i) show 5376 9833 moveto 384 /Symbol f1 (\327) show 5513 9833 moveto 384 /Symbol f2 (b) show 5774 9929 moveto 224 /Times-Italic f1 (j) show 5923 9833 moveto 384 /Symbol f1 (\327) show 6064 9833 moveto 384 /Times-Italic f1 (sin) show 6650 9833 moveto 384 /Symbol f2 (w) show 7020 9833 moveto 384 /Symbol f1 (\327) show 7176 9833 moveto 384 /Times-Italic f1 (t) show 6519 9843 moveto 384 1000 1171 /Symbol f3 (\() show 7304 9843 moveto (\)) show 16 th 2885 9441 moveto 4557 0 rlineto stroke 2686 9228 moveto 384 /Symbol f1 (\346) show 2686 9897 moveto (\350) show 2686 9678 moveto (\347) show 7454 9228 moveto (\366) show 7454 9897 moveto (\370) show 7454 9678 moveto (\367) show 7634 8968 moveto 224 /Times-Roman f1 (1) show 7756 8968 moveto 224 /Times-Roman f1 (/) show 7849 8968 moveto 224 /Times-Roman f1 (2) show end pse gR gS 190 687 215 82 rC 405 769 :M psb currentpoint pse 190 687 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6880 div 2624 3 -1 roll exch div scale currentpoint translate 64 -3610 translate -9 4272 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (d) show 223 4368 moveto 224 ns (j) show 457 4272 moveto 384 ns (t) show 314 4282 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 585 4282 moveto (\)) show 799 4272 moveto 384 /Symbol f1 (=) show 3154 3989 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (b) show 3415 4085 moveto 224 /Times-Italic f1 (j) show 3564 3989 moveto 384 /Symbol f1 (\327) show 3708 3989 moveto 384 /Symbol f2 (w) show 4034 3818 moveto 224 /Times-Roman f1 (2) show 1260 4586 moveto 384 /Symbol f2 (w) show 1586 4415 moveto 224 /Times-Roman f1 (2) show 1794 4586 moveto 384 /Symbol f1 (\327) show 1945 4586 moveto 384 /Times-Roman f1 (cos) show 2466 4415 moveto 224 /Times-Roman f1 (2) show 2747 4586 moveto 384 /Symbol f2 (w) show 3117 4586 moveto 384 /Symbol f1 (\327) show 3273 4586 moveto 384 /Times-Italic f1 (t) show 2616 4596 moveto 384 1000 1171 /Symbol f3 (\() show 3401 4596 moveto (\)) show 3600 4586 moveto 384 /Symbol f1 (+) show 3879 4586 moveto 384 /Symbol f2 (b) show 4140 4682 moveto 224 /Times-Italic f1 (j) show 4213 4420 moveto 224 /Times-Roman f1 (2) show 4421 4586 moveto 384 /Symbol f1 (\327) show 4562 4586 moveto 384 /Times-Roman f1 (sin) show 5031 4415 moveto 224 /Times-Roman f1 (2) show 5312 4586 moveto 384 /Symbol f2 (w) show 5682 4586 moveto 384 /Symbol f1 (\327) show 5838 4586 moveto 384 /Times-Italic f1 (t) show 5181 4596 moveto 384 1000 1171 /Symbol f3 (\() show 5966 4596 moveto (\)) show 1129 4641 moveto 384 1000 1661 /Symbol f3 (\() show 6093 4641 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1117 4173 moveto 5114 0 rlineto stroke 6324 4272 moveto 384 /Symbol f1 (+) show 0 5682 moveto 384 /Times-Roman f1 ( ) show 860 5682 moveto 384 /Symbol f1 (-) show 1185 5367 moveto 384 /Times-Italic f1 (i) show 1354 5367 moveto 384 /Symbol f1 (\327) show 1498 5367 moveto 384 /Symbol f2 (w) show 1868 5367 moveto 384 /Symbol f1 (\327) show 2132 5367 moveto 384 /Symbol f2 (b) show 2393 5463 moveto 224 /Times-Italic f1 (j) show 2466 5201 moveto 224 /Times-Roman f1 (2) show 2696 5367 moveto 384 /Symbol f1 (-) show 2981 5367 moveto 384 /Symbol f2 (w) show 3307 5196 moveto 224 /Times-Roman f1 (2) show 2008 5422 moveto 384 1000 1661 /Symbol f3 (\() show 3466 5422 moveto (\)) show 3642 5367 moveto 384 /Symbol f1 (\327) show 3783 5367 moveto 384 /Times-Roman f1 (sin) show 4369 5367 moveto 384 /Symbol f2 (w) show 4739 5367 moveto 384 /Symbol f1 (\327) show 4895 5367 moveto 384 /Times-Italic f1 (t) show 4238 5377 moveto 384 1000 1171 /Symbol f3 (\() show 5023 5377 moveto (\)) show 5199 5367 moveto 384 /Symbol f1 (\327) show 5350 5367 moveto 384 /Times-Roman f1 (cos) show 5988 5367 moveto 384 /Symbol f2 (w) show 6358 5367 moveto 384 /Symbol f1 (\327) show 6514 5367 moveto 384 /Times-Italic f1 (t) show 5857 5377 moveto 384 1000 1171 /Symbol f3 (\() show 6642 5377 moveto (\)) show 1556 5996 moveto 384 /Symbol f2 (w) show 1882 5825 moveto 224 /Times-Roman f1 (2) show 2090 5996 moveto 384 /Symbol f1 (\327) show 2241 5996 moveto 384 /Times-Roman f1 (cos) show 2762 5825 moveto 224 /Times-Roman f1 (2) show 3043 5996 moveto 384 /Symbol f2 (w) show 3413 5996 moveto 384 /Symbol f1 (\327) show 3569 5996 moveto 384 /Times-Italic f1 (t) show 2912 6006 moveto 384 1000 1171 /Symbol f3 (\() show 3697 6006 moveto (\)) show 3896 5996 moveto 384 /Symbol f1 (+) show 4175 5996 moveto 384 /Symbol f2 (b) show 4436 6092 moveto 224 /Times-Italic f1 (j) show 4509 5830 moveto 224 /Times-Roman f1 (2) show 4717 5996 moveto 384 /Symbol f1 (\327) show 4858 5996 moveto 384 /Times-Roman f1 (sin) show 5327 5825 moveto 224 /Times-Roman f1 (2) show 5608 5996 moveto 384 /Symbol f2 (w) show 5978 5996 moveto 384 /Symbol f1 (\327) show 6134 5996 moveto 384 /Times-Italic f1 (t) show 5477 6006 moveto 384 1000 1171 /Symbol f3 (\() show 6262 6006 moveto (\)) show 1425 6051 moveto 384 1000 1661 /Symbol f3 (\() show 6389 6051 moveto (\)) show 1161 5583 moveto 5619 0 rlineto stroke end pse endp showpage %%Page: 14 14 %%BeginPageSetup initializepage (defalco; pag: 14 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (14)S 72 110 :M -.165(As )A 88 100 18 15 rC 106 115 :M psb currentpoint pse 88 100 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 576 div 480 3 -1 roll exch div scale currentpoint translate -1062 59 translate 1123 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 1319 357 moveto 224 /Times-Roman f1 (1) show 1425 357 moveto 224 /Times-Roman f1 (,) show 1496 357 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 106 110 :M f56 sf -.081( and )A 129 100 13 15 rC 142 115 :M psb currentpoint pse 129 100 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 -801 translate -3 1121 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 208 1217 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 142 110 :M f56 sf .086 .009( commute, the above solution can be easily found by observing that a Gaussian)J 72 129 :M -.097(initial condition in the )A 179 119 27 14 rC 206 133 :M psb currentpoint pse 179 119 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 -13382 translate -6 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 13798 moveto 224 /Times-Roman f1 (1) show 313 13702 moveto 384 /Times-Roman f1 (,) show 448 13702 moveto 384 /Times-Italic f1 (x) show 657 13798 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 206 129 :M f56 sf .314 .031( plane evolves, under a harmonic potential, into a Gaussian wave)J 72 148 :M -.05(function; this wave function is, in turn, expanded as)A 102 170 387 32 rC 489 202 :M psb currentpoint pse 102 170 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12384 div 1024 3 -1 roll exch div scale currentpoint translate 64 -6926 translate -8 7534 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (exp) show 689 7534 moveto 384 /Symbol f1 (-) show 1076 7534 moveto 384 /Times-Italic f1 (d) show 1272 7630 moveto 224 /Times-Roman f1 (1) show 1462 7534 moveto 384 /Symbol f1 (\327) show 1616 7534 moveto 384 /Times-Italic f1 (x) show 1807 7630 moveto 224 /Times-Roman f1 (1) show 1921 7368 moveto (2) show 2152 7534 moveto 384 /Symbol f1 (+) show 2445 7534 moveto 384 /Times-Italic f1 (d) show 2659 7630 moveto 224 /Times-Roman f1 (2) show 2867 7534 moveto 384 /Symbol f1 (\327) show 3021 7534 moveto 384 /Times-Italic f1 (x) show 3230 7630 moveto 224 /Times-Roman f1 (2) show 3362 7368 moveto (2) show 938 7576 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1515 /Symbol f3 (\() show 3521 7576 moveto (\)) show 3824 7534 moveto 384 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -161 501 3818 7185 vb 546 7593 moveto 384 1000 1708 /Symbol f3 (\() show 4027 7593 moveto (\)) show 4241 7534 moveto 384 /Symbol f1 (=) show 4551 7534 moveto 384 /Times-Roman f1 (exp) show 5305 7534 moveto 384 /Symbol f1 (-) show 5597 7295 moveto 384 /Times-Italic f1 (d) show 5793 7391 moveto 224 /Times-Roman f1 (1) show 6006 7295 moveto 384 /Symbol f1 (+) show 6299 7295 moveto 384 /Times-Italic f1 (d) show 6513 7391 moveto 224 /Times-Roman f1 (2) show 6036 7827 moveto 384 ns (4) show 16 th 5574 7435 moveto 1109 0 rlineto stroke 6753 7534 moveto 384 /Symbol f1 (\327) show 6912 7534 moveto 384 /Times-Italic f1 (r) show 7066 7363 moveto 224 /Times-Roman f1 (2) show 5110 7357 moveto 384 /Symbol f1 (\346) show 5110 7756 moveto (\350) show 7216 7357 moveto (\366) show 7216 7756 moveto (\370) show 7458 7534 moveto (\327) show 8163 7534 moveto 384 /Times-Italic f1 (e) show 8352 7363 moveto 224 ns (i) show 8427 7363 moveto 224 /Symbol f1 (\327) show 8490 7363 moveto 224 /Times-Roman f1 (2) show 8613 7363 moveto 224 /Symbol f1 (\327) show 8680 7363 moveto 224 /Times-Italic f1 (n) show 8806 7363 moveto 224 /Symbol f1 (\327) show 8871 7363 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (J) show 9139 7534 moveto 384 /Symbol f1 (\327) show 9291 7534 moveto 384 /Times-Italic f1 (i) show 9420 7363 moveto 224 ns (n) show 9631 7534 moveto 384 /Symbol f1 (\327) show 9788 7534 moveto 384 /Times-Italic f1 (J) show 9962 7630 moveto 224 ns (n) show 7616 7907 moveto (n) show 7755 7907 moveto 224 /Symbol f1 (=) show 7905 7907 moveto (-) show 8051 7907 moveto (\245) show 7760 7132 moveto (+) show 7907 7132 moveto (\245) show 7707 7621 moveto 576 ns (\345) show 10311 7534 moveto 384 /Times-Italic f1 (i) show 10497 7295 moveto (d) show 10693 7391 moveto 224 /Times-Roman f1 (1) show 10905 7295 moveto 384 /Symbol f1 (-) show 11197 7295 moveto 384 /Times-Italic f1 (d) show 11411 7391 moveto 224 /Times-Roman f1 (2) show 10935 7827 moveto 384 ns (4) show 10474 7435 moveto 1107 0 rlineto stroke 11651 7534 moveto 384 /Symbol f1 (\327) show 11810 7534 moveto 384 /Times-Italic f1 (r) show 11964 7363 moveto 224 /Times-Roman f1 (2) show 10120 7357 moveto 384 /Symbol f1 (\346) show 10120 7756 moveto (\350) show 12114 7357 moveto (\366) show 12114 7756 moveto (\370) show end pse gR gS 17 23 560 797 rC 489 189 :M f56 sf (,)S 72 234 :M .432 .043(where, )J 108 227 7 8 rC 115 235 :M psb currentpoint pse 108 227 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 256 3 -1 roll exch div scale currentpoint translate 64 -13478 translate -1 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (r) show end pse gR gS 17 23 560 797 rC 115 234 :M f56 sf -.081( and )A 138 224 12 12 rC 150 236 :M psb currentpoint pse 138 224 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 384 div 384 3 -1 roll exch div scale currentpoint translate 64 56 translate -6 264 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (J) show end pse gR gS 17 23 560 797 rC 150 234 :M f56 sf -.134( are the polar coordinates in the plane )A 328 224 27 14 rC 355 238 :M psb currentpoint pse 328 224 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 -13382 translate -6 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 13798 moveto 224 /Times-Roman f1 (1) show 313 13702 moveto 384 /Times-Roman f1 (,) show 448 13702 moveto 384 /Times-Italic f1 (x) show 657 13798 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 355 234 :M f56 sf .048 .005(; the above expansion reduces then)J 72 251 :M .155 .015(the problem of motion in the )J 214 241 13 15 rC 227 256 :M psb currentpoint pse 214 241 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate -390 -13382 translate 448 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 654 13798 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 227 251 :M f56 sf .133 .013( direction to a sequence of elementary problems in a constant)J 72 271 :M .103 .01(force, with a Gaussian initial condition. As Figure 3 shows, for our choice of parameters only)J 72 289 :M -.099(terms with )A 125 282 10 8 rC 135 290 :M psb currentpoint pse 125 282 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -3 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (n) show end pse gR gS 17 23 560 797 rC 135 289 :M f56 sf (ranging from )S 201 279 15 12 rC 216 291 :M psb currentpoint pse 201 279 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (-) show 206 261 moveto 384 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 216 289 :M f56 sf .724 .072( to )J 233 279 15 12 rC 248 291 :M psb currentpoint pse 233 279 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -3 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (+) show 207 261 moveto 384 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 248 289 :M f56 sf .226 .023( contribute significantly to the above expansion, and are,)J 72 307 :M -.106(in fact, taken into account in our numerical simulations.)A 72 343 :M -.072(Figure 4 gives a fairly intuitive stochastic mechanical description of "space quantization" in the)A 72 349 13 15 rC 85 364 :M psb currentpoint pse 72 349 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 -6345 translate -6 6665 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 200 6761 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 85 359 :M f56 sf 1.293 .129( direction in terms of sample paths of the stochastic process )J 396 347 48 17 rC 444 364 :M psb currentpoint pse 396 347 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1536 div 544 3 -1 roll exch div scale currentpoint translate 64 -1586 translate -3 1970 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 202 2066 moveto 224 /Times-Roman f1 (3) show 485 1970 moveto 384 /Times-Italic f1 (t) show 600 1970 moveto 384 /Times-Roman f1 (,) show 732 1970 moveto 384 /Times-Italic f1 (q) show 1078 1970 moveto (t) show 935 1980 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1206 1980 moveto (\)) show 342 1998 moveto 384 1000 1364 /Symbol f3 (\() show 1333 1998 moveto (\)) show end pse gR gS 17 23 560 797 rC 444 359 :M f56 sf 1.531 .153( \(mean forward)J 72 377 :M -.177(velocity in the )A 141 367 13 15 rC 154 382 :M psb currentpoint pse 141 367 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate -390 -13382 translate 448 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 654 13798 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 154 377 :M f56 sf -.18( direction\):)A 161 401 265 163 rC 265 163 8 795 [265 0 0 163 161 401] F T 1 F :f :l 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF%ADOeod :j gR gS 17 23 560 797 rC 1 65.5 400 @w 1 65.5 400 @w 452 66 400.5 @c 1 518.5 400 @w 1 518.5 400 @w 163 65.5 401 @w 1 65.5 564 @w 1 65.5 564 @w 452 66 564.5 @c 163 518.5 401 @w 1 518.5 564 @w 1 518.5 564 @w 276 574 :M f56 sf -.046(Figure 4)A 227 578 48 17 rC 275 595 :M psb currentpoint pse 227 578 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1536 div 544 3 -1 roll exch div scale currentpoint translate 64 -1586 translate -3 1970 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 202 2066 moveto 224 /Times-Roman f1 (3) show 485 1970 moveto 384 /Times-Italic f1 (t) show 600 1970 moveto 384 /Times-Roman f1 (,) show 732 1970 moveto 384 /Times-Italic f1 (q) show 1078 1970 moveto (t) show 935 1980 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1206 1980 moveto (\)) show 342 1998 moveto 384 1000 1364 /Symbol f3 (\() show 1333 1998 moveto (\)) show end pse gR gS 17 23 560 797 rC 275 590 :M f193 sf -.078( as a function of )A 354 582 7 10 rC 361 592 :M psb currentpoint pse 354 582 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -606 -36 translate 666 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 361 590 :M f193 sf (.)S 72 627 :M f56 sf -.023(Figure 2 follows, in an analogous way, the motion in the )A 345 617 27 14 rC 372 631 :M psb currentpoint pse 345 617 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 -13382 translate -6 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 13798 moveto 224 /Times-Roman f1 (1) show 313 13702 moveto 384 /Times-Roman f1 (,) show 448 13702 moveto 384 /Times-Italic f1 (x) show 657 13798 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 372 627 :M f56 sf .3 .03( plane. In order to motivate the)J 72 646 :M .455 .046(introduction of the quantity plotted there, we recall, first of all, an elementary fact about the)J 72 662 :M -.122(operator )A 114 652 14 15 rC 128 667 :M psb currentpoint pse 114 652 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 -1682 translate -4 2002 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 2098 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 128 662 :M f56 sf -.165(: if )A 144 650 150 17 rC 294 667 :M psb currentpoint pse 144 650 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4800 div 544 3 -1 roll exch div scale currentpoint translate 64 -2595 translate -4 2979 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 3075 moveto 224 /Times-Italic f1 (t) show 559 2979 moveto 384 ns (x) show 418 2989 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 763 2989 moveto (\)) show 977 2979 moveto 384 /Symbol f1 (=) show 1287 2979 moveto 384 /Times-Roman f1 (exp) show 1984 2979 moveto 384 /Times-Italic f1 (R) show 2390 2979 moveto (t) show 2505 2979 moveto 384 /Times-Roman f1 (,) show 2640 2979 moveto 384 /Times-Italic f1 (x) show 2247 2989 moveto 384 1000 1171 /Symbol f3 (\() show 2844 2989 moveto (\)) show 3043 2979 moveto 384 /Symbol f1 (+) show 3337 2979 moveto 384 /Times-Italic f1 (i) show 3506 2979 moveto 384 /Symbol f1 (\327) show 3650 2979 moveto 384 /Times-Italic f1 (S) show 3993 2979 moveto (t) show 4108 2979 moveto 384 /Times-Roman f1 (,) show 4243 2979 moveto 384 /Times-Italic f1 (x) show 3850 2989 moveto 384 1000 1171 /Symbol f3 (\() show 4447 2989 moveto (\)) show 1841 3007 moveto 384 1000 1364 /Symbol f3 (\() show 4574 3007 moveto (\)) show end pse gR gS 17 23 560 797 rC 294 662 :M f56 sf .257 .026(, with )J 325 651 34 15 rC 359 666 :M psb currentpoint pse 325 651 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 480 3 -1 roll exch div scale currentpoint translate -1924 -2627 translate 1984 2979 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (R) show 2390 2979 moveto (t) show 2505 2979 moveto 384 /Times-Roman f1 (,) show 2640 2979 moveto 384 /Times-Italic f1 (x) show 2247 2989 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 2844 2989 moveto (\)) show end pse gR gS 17 23 560 797 rC 359 662 :M f56 sf -.081( and )A 382 651 31 15 rC 413 666 :M psb currentpoint pse 382 651 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 992 div 480 3 -1 roll exch div scale currentpoint translate -3602 -2627 translate 3650 2979 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (S) show 3993 2979 moveto (t) show 4108 2979 moveto 384 /Times-Roman f1 (,) show 4243 2979 moveto 384 /Times-Italic f1 (x) show 3850 2989 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 4447 2989 moveto (\)) show end pse gR gS 17 23 560 797 rC 413 662 :M f56 sf -.065( real functions, then)A 122 686 350 36 rC 472 722 :M psb currentpoint pse 122 686 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11200 div 1152 3 -1 roll exch div scale currentpoint translate 64 -3285 translate 370 3718 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (1) show 28 4250 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 344 4346 moveto 224 /Times-Italic f1 (t) show 591 4250 moveto 384 ns (x) show 450 4260 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 795 4260 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 3858 moveto 933 0 rlineto stroke 1003 3957 moveto 384 /Symbol f1 (\327) show 1159 3957 moveto 384 /Times-Italic f1 (L) show 1411 4053 moveto 224 /Times-Roman f1 (3) show 1535 3957 moveto 384 /Symbol f2 (y) show 1851 4053 moveto 224 /Times-Italic f1 (t) show 2098 3957 moveto 384 ns (x) show 1957 3967 moveto 384 1000 1171 /Symbol f3 (\() show 2302 3967 moveto (\)) show 2516 3957 moveto 384 /Symbol f1 (=) show 2828 3957 moveto 384 /Times-Italic f1 (x) show 3019 4053 moveto 224 /Times-Roman f1 (1) show 3226 3718 moveto 384 /Symbol f2 (\266) show 3446 3718 moveto 384 /Times-Italic f1 (S) show 3789 3718 moveto (t) show 3904 3718 moveto 384 /Times-Roman f1 (,) show 4039 3718 moveto 384 /Times-Italic f1 (x) show 3646 3728 moveto 384 1000 1171 /Symbol f3 (\() show 4243 3728 moveto (\)) show 3504 4250 moveto 384 /Symbol f2 (\266) show 3724 4250 moveto 384 /Times-Italic f1 (x) show 3933 4346 moveto 224 /Times-Roman f1 (2) show 3203 3858 moveto 1178 0 rlineto stroke 4473 3957 moveto 384 /Symbol f1 (-) show 4768 3957 moveto 384 /Times-Italic f1 (x) show 4977 4053 moveto 224 /Times-Roman f1 (2) show 5202 3718 moveto 384 /Symbol f2 (\266) show 5422 3718 moveto 384 /Times-Italic f1 (S) show 5765 3718 moveto (t) show 5880 3718 moveto 384 /Times-Roman f1 (,) show 6015 3718 moveto 384 /Times-Italic f1 (x) show 5622 3728 moveto 384 1000 1171 /Symbol f3 (\() show 6219 3728 moveto (\)) show 5498 4250 moveto 384 /Symbol f2 (\266) show 5718 4250 moveto 384 /Times-Italic f1 (x) show 5909 4346 moveto 224 /Times-Roman f1 (1) show 5179 3858 moveto 1178 0 rlineto stroke 6449 3957 moveto 384 /Symbol f1 (-) show 6742 3957 moveto 384 /Times-Italic f1 (i) show 6911 3957 moveto 384 /Symbol f1 (\327) show 7249 3957 moveto 384 /Times-Italic f1 (x) show 7440 4053 moveto 224 /Times-Roman f1 (1) show 7647 3718 moveto 384 /Symbol f2 (\266) show 7867 3718 moveto 384 /Times-Italic f1 (R) show 8273 3718 moveto (t) show 8388 3718 moveto 384 /Times-Roman f1 (,) show 8523 3718 moveto 384 /Times-Italic f1 (x) show 8130 3728 moveto 384 1000 1171 /Symbol f3 (\() show 8727 3728 moveto (\)) show 7974 4250 moveto 384 /Symbol f2 (\266) show 8194 4250 moveto 384 /Times-Italic f1 (x) show 8385 4346 moveto 224 /Times-Roman f1 (1) show 7624 3858 moveto 1241 0 rlineto stroke 8957 3957 moveto 384 /Symbol f1 (-) show 9252 3957 moveto 384 /Times-Italic f1 (x) show 9461 4053 moveto 224 /Times-Roman f1 (2) show 9686 3718 moveto 384 /Symbol f2 (\266) show 9906 3718 moveto 384 /Times-Italic f1 (R) show 10312 3718 moveto (t) show 10427 3718 moveto 384 /Times-Roman f1 (,) show 10562 3718 moveto 384 /Times-Italic f1 (x) show 10169 3728 moveto 384 1000 1171 /Symbol f3 (\() show 10766 3728 moveto (\)) show 9995 4250 moveto 384 /Symbol f2 (\266) show 10215 4250 moveto 384 /Times-Italic f1 (x) show 10424 4346 moveto 224 /Times-Roman f1 (2) show 9663 3858 moveto 1241 0 rlineto stroke 7056 3693 moveto 384 /Symbol f1 (\346) show 7056 4266 moveto (\350) show 7056 4047 moveto (\347) show 10916 3693 moveto (\366) show 10916 4266 moveto (\370) show 10916 4047 moveto (\367) show end pse gR gS 17 23 560 797 rC 72 755 :M f56 sf -.033(If, therefore, at some point )A 202 748 9 8 rC 211 756 :M psb currentpoint pse 202 748 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show end pse gR gS 17 23 560 797 rC 211 755 :M f56 sf (, it is, for some real )S 307 748 13 8 rC 320 756 :M psb currentpoint pse 307 748 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 256 3 -1 roll exch div scale currentpoint translate 64 -37 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show end pse gR gS 17 23 560 797 rC 320 755 :M f56 sf (,)S endp showpage %%Page: 15 15 %%BeginPageSetup initializepage (defalco; pag: 15 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (15)S 72 115 :M ( )S 249 98 101 31 rC 350 129 :M psb currentpoint pse 249 98 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3232 div 992 3 -1 roll exch div scale currentpoint translate 64 -75 translate 370 380 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (1) show 28 912 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 344 1008 moveto 224 /Times-Italic f1 (t) show 591 912 moveto 384 ns (x) show 450 922 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 795 922 moveto (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 520 moveto 933 0 rlineto stroke 1003 619 moveto 384 /Symbol f1 (\327) show 1159 619 moveto 384 /Times-Italic f1 (L) show 1411 715 moveto 224 /Times-Roman f1 (3) show 1535 619 moveto 384 /Symbol f2 (y) show 1851 715 moveto 224 /Times-Italic f1 (t) show 2098 619 moveto 384 ns (x) show 1957 629 moveto 384 1000 1171 /Symbol f3 (\() show 2302 629 moveto (\)) show 2516 619 moveto 384 /Symbol f1 (=) show 2830 619 moveto 384 /Times-Italic f1 (m) show end pse gR gS 17 23 560 797 rC 350 115 :M f56 sf .019 .002(, \(13\))J 72 162 :M -.118(it follows that)A 229 166 136 32 rC 365 198 :M psb currentpoint pse 229 166 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4352 div 1024 3 -1 roll exch div scale currentpoint translate 64 45 translate -6 531 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 627 moveto 224 /Times-Roman f1 (1) show 392 292 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 612 292 moveto 384 /Times-Italic f1 (S) show 955 292 moveto (t) show 1070 292 moveto 384 /Times-Roman f1 (,) show 1205 292 moveto 384 /Times-Italic f1 (x) show 812 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1409 302 moveto (\)) show 670 824 moveto 384 /Symbol f2 (\266) show 890 824 moveto 384 /Times-Italic f1 (x) show 1099 920 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 369 432 moveto 1178 0 rlineto stroke 1639 531 moveto 384 /Symbol f1 (-) show 1934 531 moveto 384 /Times-Italic f1 (x) show 2143 627 moveto 224 /Times-Roman f1 (2) show 2368 292 moveto 384 /Symbol f2 (\266) show 2588 292 moveto 384 /Times-Italic f1 (S) show 2931 292 moveto (t) show 3046 292 moveto 384 /Times-Roman f1 (,) show 3181 292 moveto 384 /Times-Italic f1 (x) show 2788 302 moveto 384 1000 1171 /Symbol f3 (\() show 3385 302 moveto (\)) show 2664 824 moveto 384 /Symbol f2 (\266) show 2884 824 moveto 384 /Times-Italic f1 (x) show 3075 920 moveto 224 /Times-Roman f1 (1) show 2345 432 moveto 1178 0 rlineto stroke 3631 531 moveto 384 /Symbol f1 (=) show 3945 531 moveto 384 /Times-Italic f1 (m) show end pse gR gS 17 23 560 797 rC 72 213 :M f56 sf -.163(and)A 223 217 148 36 rC 371 253 :M psb currentpoint pse 223 217 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4736 div 1152 3 -1 roll exch div scale currentpoint translate 64 -1099 translate 178 1771 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 369 1867 moveto 224 /Times-Roman f1 (1) show 576 1532 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (\266) show 796 1532 moveto 384 /Times-Italic f1 (R) show 1202 1532 moveto (t) show 1317 1532 moveto 384 /Times-Roman f1 (,) show 1452 1532 moveto 384 /Times-Italic f1 (x) show 1059 1542 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1656 1542 moveto (\)) show 903 2064 moveto 384 /Symbol f2 (\266) show 1123 2064 moveto 384 /Times-Italic f1 (x) show 1314 2160 moveto 224 /Times-Roman f1 (1) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 553 1672 moveto 1241 0 rlineto stroke 1886 1771 moveto 384 /Symbol f1 (-) show 2181 1771 moveto 384 /Times-Italic f1 (x) show 2390 1867 moveto 224 /Times-Roman f1 (2) show 2615 1532 moveto 384 /Symbol f2 (\266) show 2835 1532 moveto 384 /Times-Italic f1 (R) show 3241 1532 moveto (t) show 3356 1532 moveto 384 /Times-Roman f1 (,) show 3491 1532 moveto 384 /Times-Italic f1 (x) show 3098 1542 moveto 384 1000 1171 /Symbol f3 (\() show 3695 1542 moveto (\)) show 2924 2064 moveto 384 /Symbol f2 (\266) show 3144 2064 moveto 384 /Times-Italic f1 (x) show 3353 2160 moveto 224 /Times-Roman f1 (2) show 2592 1672 moveto 1241 0 rlineto stroke -15 1507 moveto 384 /Symbol f1 (\346) show -15 2080 moveto (\350) show -15 1861 moveto (\347) show 3845 1507 moveto (\366) show 3845 2080 moveto (\370) show 3845 1861 moveto (\367) show 4125 1771 moveto (=) show 4434 1771 moveto 384 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 72 284 :M f56 sf -.235(We recall that )A 139 272 124 17 rC 263 289 :M psb currentpoint pse 139 272 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3968 div 544 3 -1 roll exch div scale currentpoint translate 32 -6387 translate 29 6771 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (u) show 367 6771 moveto (t) show 482 6771 moveto 384 /Times-Roman f1 (,) show 614 6771 moveto 384 /Times-Italic f1 (q) show 960 6771 moveto (t) show 817 6781 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1088 6781 moveto (\)) show 224 6799 moveto 384 1000 1364 /Symbol f3 (\() show 1215 6799 moveto (\)) show 1429 6771 moveto 384 /Symbol f1 (=) show 1737 6771 moveto 384 /Times-Italic f1 (grad) show 2418 6771 moveto 384 /Times-Roman f1 ( ) show 2514 6771 moveto 384 /Times-Italic f1 (R) show 2920 6771 moveto (t) show 3035 6771 moveto 384 /Times-Roman f1 (,) show 3167 6771 moveto 384 /Times-Italic f1 (q) show 3513 6771 moveto (t) show 3370 6781 moveto 384 1000 1171 /Symbol f3 (\() show 3641 6781 moveto (\)) show 2777 6799 moveto 384 1000 1364 /Symbol f3 (\() show 3768 6799 moveto (\)) show end pse gR gS 17 23 560 797 rC 263 284 :M f56 sf .208 .021( has, in stochastic mechanics, the meaning of osmotic)J 72 302 :M .242 .024("velocity", while )J 157 290 177 17 rC 334 307 :M psb currentpoint pse 157 290 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5664 div 544 3 -1 roll exch div scale currentpoint translate 32 -5704 translate 29 6088 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (v) show 363 6088 moveto (t) show 478 6088 moveto 384 /Times-Roman f1 (,) show 610 6088 moveto 384 /Times-Italic f1 (q) show 956 6088 moveto (t) show 813 6098 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1084 6098 moveto (\)) show 220 6116 moveto 384 1000 1364 /Symbol f3 (\() show 1211 6116 moveto (\)) show 1425 6088 moveto 384 /Symbol f1 (=) show 1733 6088 moveto 384 /Times-Italic f1 (grad) show 2414 6088 moveto 384 /Times-Roman f1 ( ) show 2510 6088 moveto 384 /Times-Italic f1 (S) show 2853 6088 moveto (t) show 2968 6088 moveto 384 /Times-Roman f1 (,) show 3100 6088 moveto 384 /Times-Italic f1 (q) show 3446 6088 moveto (t) show 3303 6098 moveto 384 1000 1171 /Symbol f3 (\() show 3574 6098 moveto (\)) show 2710 6116 moveto 384 1000 1364 /Symbol f3 (\() show 3701 6116 moveto (\)) show 3899 6088 moveto 384 /Symbol f1 (-) show 4195 6088 moveto 384 /Times-Italic f1 (A) show 4621 6088 moveto (t) show 4736 6088 moveto 384 /Times-Roman f1 (,) show 4868 6088 moveto 384 /Times-Italic f1 (q) show 5214 6088 moveto (t) show 5071 6098 moveto 384 1000 1171 /Symbol f3 (\() show 5342 6098 moveto (\)) show 4478 6116 moveto 384 1000 1364 /Symbol f3 (\() show 5469 6116 moveto (\)) show end pse gR gS 17 23 560 797 rC 334 302 :M f56 sf .282 .028( has the meaning of current velocity. It)J 72 322 :M -.005(is therefore natural, in perfect analogy with the classical canonical formalism, to introduce the)A 72 338 :M -.16(canonical momentum random variables )A 259 326 230 17 rC 489 343 :M psb currentpoint pse 259 326 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7360 div 544 3 -1 roll exch div scale currentpoint translate 32 -6387 translate 32 6771 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 367 6771 moveto (t) show 482 6771 moveto 384 /Times-Roman f1 (,) show 614 6771 moveto 384 /Times-Italic f1 (q) show 960 6771 moveto (t) show 817 6781 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1088 6781 moveto (\)) show 224 6799 moveto 384 1000 1364 /Symbol f3 (\() show 1215 6799 moveto (\)) show 1429 6771 moveto 384 /Symbol f1 (=) show 1744 6771 moveto 384 /Times-Italic f1 (v) show 2078 6771 moveto (t) show 2193 6771 moveto 384 /Times-Roman f1 (,) show 2325 6771 moveto 384 /Times-Italic f1 (q) show 2671 6771 moveto (t) show 2528 6781 moveto 384 1000 1171 /Symbol f3 (\() show 2799 6781 moveto (\)) show 1935 6799 moveto 384 1000 1364 /Symbol f3 (\() show 2926 6799 moveto (\)) show 3125 6771 moveto 384 /Symbol f1 (+) show 3422 6771 moveto 384 /Times-Italic f1 (A) show 3848 6771 moveto (t) show 3963 6771 moveto 384 /Times-Roman f1 (,) show 4095 6771 moveto 384 /Times-Italic f1 (q) show 4441 6771 moveto (t) show 4298 6781 moveto 384 1000 1171 /Symbol f3 (\() show 4569 6781 moveto (\)) show 3705 6799 moveto 384 1000 1364 /Symbol f3 (\() show 4696 6799 moveto (\)) show 4910 6771 moveto 384 /Symbol f1 (=) show 5218 6771 moveto 384 /Times-Italic f1 (grad) show 5899 6771 moveto 384 /Times-Roman f1 ( ) show 5995 6771 moveto 384 /Times-Italic f1 (S) show 6338 6771 moveto (t) show 6453 6771 moveto 384 /Times-Roman f1 (,) show 6585 6771 moveto 384 /Times-Italic f1 (q) show 6931 6771 moveto (t) show 6788 6781 moveto 384 1000 1171 /Symbol f3 (\() show 7059 6781 moveto (\)) show 6195 6799 moveto 384 1000 1364 /Symbol f3 (\() show 7186 6799 moveto (\)) show end pse gR gS 17 23 560 797 rC 489 338 :M f56 sf .515 .052( and to)J 72 356 :M (define )S 105 344 224 17 rC 329 361 :M psb currentpoint pse 105 344 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7168 div 544 3 -1 roll exch div scale currentpoint translate 32 -7070 translate 23 7454 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) show 275 7550 moveto 224 /Times-Roman f1 (3) show 558 7454 moveto 384 /Times-Italic f1 (t) show 673 7454 moveto 384 /Times-Roman f1 (,) show 805 7454 moveto 384 /Times-Italic f1 (q) show 1151 7454 moveto (t) show 1008 7464 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1279 7464 moveto (\)) show 415 7482 moveto 384 1000 1364 /Symbol f3 (\() show 1406 7482 moveto (\)) show 1619 7454 moveto 384 /Symbol f1 (\272) show 1928 7454 moveto 384 /Times-Italic f1 (q) show 2127 7550 moveto 224 /Times-Roman f1 (1) show 2402 7454 moveto 384 /Times-Italic f1 (t) show 2259 7464 moveto 384 1000 1171 /Symbol f3 (\() show 2530 7464 moveto (\)) show 2706 7454 moveto 384 /Symbol f1 (\327) show 2866 7454 moveto 384 /Times-Italic f1 (p) show 3072 7550 moveto 224 /Times-Roman f1 (2) show 3365 7454 moveto 384 /Times-Italic f1 (t) show 3480 7454 moveto 384 /Times-Roman f1 (,) show 3612 7454 moveto 384 /Times-Italic f1 (q) show 3958 7454 moveto (t) show 3815 7464 moveto 384 1000 1171 /Symbol f3 (\() show 4086 7464 moveto (\)) show 3222 7482 moveto 384 1000 1364 /Symbol f3 (\() show 4213 7482 moveto (\)) show 4411 7454 moveto 384 /Symbol f1 (-) show 4703 7454 moveto 384 /Times-Italic f1 (q) show 4920 7550 moveto 224 /Times-Roman f1 (2) show 5213 7454 moveto 384 /Times-Italic f1 (t) show 5070 7464 moveto 384 1000 1171 /Symbol f3 (\() show 5341 7464 moveto (\)) show 5517 7454 moveto 384 /Symbol f1 (\327) show 5677 7454 moveto 384 /Times-Italic f1 (p) show 5865 7550 moveto 224 /Times-Roman f1 (1) show 6140 7454 moveto 384 /Times-Italic f1 (t) show 6255 7454 moveto 384 /Times-Roman f1 (,) show 6387 7454 moveto 384 /Times-Italic f1 (q) show 6733 7454 moveto (t) show 6590 7464 moveto 384 1000 1171 /Symbol f3 (\() show 6861 7464 moveto (\)) show 5997 7482 moveto 384 1000 1364 /Symbol f3 (\() show 6988 7482 moveto (\)) show end pse gR gS 17 23 560 797 rC 329 356 :M f56 sf .48 .048( as the stochastic analogue of canonical)J 72 376 :M -.112(angular momentum \(Busch constant of motion for a classical system moving in a magnetic field)A 72 392 :M -.122(with rotational symmetry around the )A 246 382 13 15 rC 259 397 :M psb currentpoint pse 246 382 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 32 -7754 translate 26 8074 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 232 8170 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 259 392 :M f56 sf .425 .042( axis\).)J 72 410 :M .558 .056(As )J 89 402 7 10 rC 96 412 :M psb currentpoint pse 89 402 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate 64 -1157 translate -4 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 96 410 :M f56 sf .43 .043( increases, the components of )J 244 400 15 15 rC 259 415 :M psb currentpoint pse 244 400 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 480 3 -1 roll exch div scale currentpoint translate 64 -1093 translate -4 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 1509 moveto 224 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 259 410 :M f56 sf .674 .067( belonging to different eigenvalues )J 435 403 13 8 rC 448 411 :M psb currentpoint pse 435 403 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 256 3 -1 roll exch div scale currentpoint translate 64 -1189 translate -4 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show end pse gR gS 17 23 560 797 rC 448 410 :M f56 sf 1.742 .174( of )J 468 400 14 15 rC 482 415 :M psb currentpoint pse 468 400 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 -1682 translate -4 2002 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 2098 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 482 410 :M f56 sf .43 .043( become)J 72 428 :M .158 .016(sharply separated along the )J 207 418 13 15 rC 220 433 :M psb currentpoint pse 207 418 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 -1093 translate -6 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 200 1509 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 220 428 :M f56 sf .361 .036( direction; this fact has the consequence \(because, for large )J 512 420 7 10 rC 519 430 :M psb currentpoint pse 512 420 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate 64 -1157 translate -4 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 519 428 :M f56 sf (,)S 72 448 :M .332 .033(\(13\) is, to a high degree of approximation, locally satisfied\) that the contribution of osmotic)J 72 466 :M .19 .019(velocity to angular momentum becomes negligible, while the contribution of current velocity)J 72 484 :M .793 .079(tends to a discrete random variable: this is precisely what is shown by the sample paths of)J 72 488 50 17 rC 122 505 :M psb currentpoint pse 72 488 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1600 div 544 3 -1 roll exch div scale currentpoint translate 32 -7070 translate 23 7454 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (l) show 275 7550 moveto 224 /Times-Roman f1 (3) show 558 7454 moveto 384 /Times-Italic f1 (t) show 673 7454 moveto 384 /Times-Roman f1 (,) show 805 7454 moveto 384 /Times-Italic f1 (q) show 1151 7454 moveto (t) show 1008 7464 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1279 7464 moveto (\)) show 415 7482 moveto 384 1000 1364 /Symbol f3 (\() show 1406 7482 moveto (\)) show end pse gR gS 17 23 560 797 rC 122 500 :M f56 sf .162 .016( plotted in Figure 2. Figure 2' follows, for completeness, the moment, with respect)J 72 518 :M -.166(to the )A 101 508 13 15 rC 114 523 :M psb currentpoint pse 101 508 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 32 -7754 translate 26 8074 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 232 8170 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 114 518 :M f56 sf -.027( axis, of osmotic "velocity".)A 161 542 265 163 rC 265 163 8 795 [265 0 0 163 161 542] F T 1 F :f :l 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dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 190 452 moveto 224 /Times-Roman f1 (1) show 465 356 moveto 384 /Times-Italic f1 (t) show 322 366 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 593 366 moveto (\)) show 769 356 moveto 384 /Symbol f1 (\327) show 1053 356 moveto 384 /Times-Italic f1 (b) show 1261 452 moveto 224 /Times-Roman f1 (2) show 1554 356 moveto 384 /Times-Italic f1 (t) show 1669 356 moveto 384 /Times-Roman f1 (,) show 1801 356 moveto 384 /Times-Italic f1 (q) show 2147 356 moveto (t) show 2004 366 moveto 384 1000 1171 /Symbol f3 (\() show 2275 366 moveto (\)) show 1411 384 moveto 384 1000 1364 /Symbol f3 (\() show 2402 384 moveto (\)) show 2601 356 moveto 384 /Symbol f1 (+) show 2898 356 moveto 384 /Times-Italic f1 (A) show 3195 452 moveto 224 /Times-Roman f1 (2) show 3483 356 moveto 384 /Times-Italic f1 (q) show 3829 356 moveto (t) show 3686 366 moveto 384 1000 1171 /Symbol f3 (\() show 3957 366 moveto (\)) show 3345 384 moveto 384 1000 1364 /Symbol f3 (\() show 4084 384 moveto (\)) show 909 401 moveto 384 1000 1557 /Symbol f3 (\() show 4211 401 moveto (\)) show 4409 356 moveto 384 /Symbol f1 (-) show 4701 356 moveto 384 /Times-Italic f1 (q) show 4918 452 moveto 224 /Times-Roman f1 (2) show 5211 356 moveto 384 /Times-Italic f1 (t) show 5068 366 moveto 384 1000 1171 /Symbol f3 (\() show 5339 366 moveto (\)) show 5515 356 moveto 384 /Symbol f1 (\327) show 5799 356 moveto 384 /Times-Italic f1 (b) show 5989 452 moveto 224 /Times-Roman f1 (1) show 6264 356 moveto 384 /Times-Italic f1 (t) show 6379 356 moveto 384 /Times-Roman f1 (,) show 6511 356 moveto 384 /Times-Italic f1 (q) show 6857 356 moveto (t) show 6714 366 moveto 384 1000 1171 /Symbol f3 (\() show 6985 366 moveto (\)) show 6121 384 moveto 384 1000 1364 /Symbol f3 (\() show 7112 384 moveto (\)) show 7311 356 moveto 384 /Symbol f1 (+) show 7608 356 moveto 384 /Times-Italic f1 (A) show 7887 452 moveto 224 /Times-Roman f1 (1) show 8157 356 moveto 384 /Times-Italic f1 (q) show 8503 356 moveto (t) show 8360 366 moveto 384 1000 1171 /Symbol f3 (\() show 8631 366 moveto (\)) show 8019 384 moveto 384 1000 1364 /Symbol f3 (\() show 8758 384 moveto (\)) show 5655 401 moveto 384 1000 1557 /Symbol f3 (\() show 8885 401 moveto (\)) show end pse gR gS 17 23 560 797 rC 393 342 :M f193 sf -.078( as a function of )A 472 334 7 10 rC 479 344 :M psb currentpoint pse 472 334 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -606 -36 translate 666 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 479 342 :M f193 sf (.)S 72 399 :M f67 sf -.036(4. Discussion)A 72 417 :M f56 sf -.127(We summarise the considerations made up to this point into the statements:)A 72 457 :M ( )S 234 439 109 31 rC 343 470 :M psb currentpoint pse 234 439 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3488 div 992 3 -1 roll exch div scale currentpoint translate 64 45 translate -4 531 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 627 moveto 224 /Times-Roman f1 (3) show 531 531 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 627 moveto 224 /Times-Roman f1 (0) show 388 561 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 561 moveto (\)) show 1217 531 moveto 384 /Symbol f1 (=) show 1621 531 moveto 384 /Times-Roman f1 (lim) show 1533 755 moveto 224 /Times-Italic f1 (t) show 1625 755 moveto 224 /Symbol f1 (-) show 1771 755 moveto (>) show 1918 755 moveto (+) show 2065 755 moveto (\245) show 2452 292 moveto 384 /Times-Italic f1 (q) show 2666 388 moveto 224 /Times-Roman f1 (3) show 2949 292 moveto 384 /Times-Italic f1 (t) show 2806 302 moveto 384 1000 1171 /Symbol f3 (\() show 3077 302 moveto (\)) show 2307 824 moveto 384 /Symbol f2 (e) show 2578 824 moveto 384 /Symbol f1 (\327) show 2734 824 moveto 384 /Times-Italic f1 (t) show 2867 653 moveto 224 /Times-Roman f1 (2) show 3148 824 moveto 384 ns (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 3141 501 vb 16 th 2283 432 moveto 1079 0 rlineto stroke end pse gR gS 17 23 560 797 rC 343 457 :M f56 sf ( \(14\))S 72 502 :M ( )S 117 489 367 22 rC 484 511 :M psb currentpoint pse 117 489 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11744 div 704 3 -1 roll exch div scale currentpoint translate 64 -2413 translate -4 2829 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 2925 moveto 224 /Times-Roman f1 (3) show 531 2829 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 2925 moveto 224 /Times-Roman f1 (0) show 388 2859 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 2859 moveto (\)) show 1217 2829 moveto 384 /Symbol f1 (=) show 1621 2829 moveto 384 /Times-Roman f1 (lim) show 1533 3053 moveto 224 /Times-Italic f1 (t) show 1625 3053 moveto 224 /Symbol f1 (-) show 1771 3053 moveto (>) show 1918 3053 moveto (+) show 2065 3053 moveto (\245) show 2274 2829 moveto 384 /Times-Italic f1 (q) show 2473 2925 moveto 224 /Times-Roman f1 (1) show 2748 2829 moveto 384 /Times-Italic f1 (t) show 2605 2839 moveto 384 1000 1171 /Symbol f3 (\() show 2876 2839 moveto (\)) show 3052 2829 moveto 384 /Symbol f1 (\327) show 3209 2829 moveto 384 /Times-Italic f1 (b) show 3417 2925 moveto 224 /Times-Roman f1 (2) show 3710 2829 moveto 384 /Times-Italic f1 (t) show 3825 2829 moveto 384 /Times-Roman f1 (,) show 3957 2829 moveto 384 /Times-Italic f1 (q) show 4303 2829 moveto (t) show 4160 2839 moveto 384 1000 1171 /Symbol f3 (\() show 4431 2839 moveto (\)) show 3567 2857 moveto 384 1000 1364 /Symbol f3 (\() show 4558 2857 moveto (\)) show 4756 2829 moveto 384 /Symbol f1 (-) show 5048 2829 moveto 384 /Times-Italic f1 (q) show 5265 2925 moveto 224 /Times-Roman f1 (2) show 5558 2829 moveto 384 /Times-Italic f1 (t) show 5415 2839 moveto 384 1000 1171 /Symbol f3 (\() show 5686 2839 moveto (\)) show 5862 2829 moveto 384 /Symbol f1 (\327) show 6019 2829 moveto 384 /Times-Italic f1 (b) show 6209 2925 moveto 224 /Times-Roman f1 (1) show 6484 2829 moveto 384 /Times-Italic f1 (t) show 6599 2829 moveto 384 /Times-Roman f1 (,) show 6731 2829 moveto 384 /Times-Italic f1 (q) show 7077 2829 moveto (t) show 6934 2839 moveto 384 1000 1171 /Symbol f3 (\() show 7205 2839 moveto (\)) show 6341 2857 moveto 384 1000 1364 /Symbol f3 (\() show 7332 2857 moveto (\)) show 7531 2829 moveto 384 /Symbol f1 (+) show 7825 2829 moveto 384 /Symbol f2 (e) show 8096 2829 moveto 384 /Symbol f1 (\327) show 8247 2829 moveto 384 /Times-Italic f1 (q) show 8461 2925 moveto 224 /Times-Roman f1 (3) show 8744 2829 moveto 384 /Times-Italic f1 (t) show 8601 2839 moveto 384 1000 1171 /Symbol f3 (\() show 8872 2839 moveto (\)) show 9048 2829 moveto 384 /Symbol f1 (\327) show 9326 2829 moveto 384 /Times-Italic f1 (q) show 9525 2925 moveto 224 /Times-Roman f1 (1) show 9800 2829 moveto 384 /Times-Italic f1 (t) show 9657 2839 moveto 384 1000 1171 /Symbol f3 (\() show 9928 2839 moveto (\)) show 10060 2632 moveto 224 /Times-Roman f1 (2) show 10291 2829 moveto 384 /Symbol f1 (+) show 10584 2829 moveto 384 /Times-Italic f1 (q) show 10801 2925 moveto 224 /Times-Roman f1 (2) show 11094 2829 moveto 384 /Times-Italic f1 (t) show 10951 2839 moveto 384 1000 1171 /Symbol f3 (\() show 11222 2839 moveto (\)) show 11354 2632 moveto 224 /Times-Roman f1 (2) show 9188 2888 moveto 384 1000 1703 /Symbol f3 (\() show 11513 2888 moveto (\)) show end pse gR gS 17 23 560 797 rC 484 502 :M f56 sf ( \(15\))S 72 544 :M -.098(where both limits are to be taken in law.)A 72 580 :M -.066(Statement \(14\) is just a rephrasing of Feynman's assertion reported in the Introduction: here the)A 72 596 :M -.164(coordinate )A 124 586 13 15 rC 137 601 :M psb currentpoint pse 124 586 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -9 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 205 357 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 137 596 :M f56 sf .347 .035( at a fixed, large, value of )J 266 588 7 10 rC 273 598 :M psb currentpoint pse 266 588 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate 64 -5 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 273 596 :M f56 sf .175 .017( plays the role of "needle" giving information about)J 72 615 :M .113 .011(the rotational state of motion in the plane )J 274 605 27 14 rC 301 619 :M psb currentpoint pse 274 605 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 357 moveto 224 /Times-Roman f1 (1) show 313 261 moveto 384 /Times-Roman f1 (,) show 448 261 moveto 384 /Times-Italic f1 (x) show 657 357 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 301 615 :M f56 sf 1.047 .105( of the "system" that one would, classically,)J 72 633 :M -.133(describe by the canonical coordinates )A 250 623 57 14 rC 307 637 :M psb currentpoint pse 250 623 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1824 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -9 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 190 357 moveto 224 /Times-Roman f1 (1) show 318 261 moveto 384 /Times-Roman f1 (,) show 459 261 moveto 384 /Times-Italic f1 (p) show 647 357 moveto 224 /Times-Roman f1 (1) show 775 261 moveto 384 /Times-Roman f1 (,) show 907 261 moveto 384 /Times-Italic f1 (q) show 1124 357 moveto 224 /Times-Roman f1 (2) show 1270 261 moveto 384 /Times-Roman f1 (,) show 1411 261 moveto 384 /Times-Italic f1 (p) show 1617 357 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 307 633 :M f56 sf (.)S 72 670 :M -.073(Statement \(15\) refers to the effects of the measuring procedure on the "system" itself. We have)A 72 688 :M .321 .032(been careful, at the end of the previous Section, to stress the fact that \(15\), though expressed)J 72 706 :M -.04(here in the notational scheme of stochastic mechanics, is just a consequence of the fact that the)A 72 722 :M -.075(components of )A 145 712 15 15 rC 160 727 :M psb currentpoint pse 145 712 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 480 3 -1 roll exch div scale currentpoint translate 64 -1093 translate -4 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 312 1509 moveto 224 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 160 722 :M f56 sf -.112( belonging to different eigenvalues of )A 339 712 14 15 rC 353 727 :M psb currentpoint pse 339 712 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 -1682 translate -4 2002 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 2098 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 353 722 :M f56 sf -.125( become sharply separated along the)A 72 730 13 15 rC 85 745 :M psb currentpoint pse 72 730 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate 64 -1093 translate -6 1413 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 200 1509 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 85 740 :M f56 sf .082 .008( direction. It is, nevertheless, interesting to observe that \(15\) - as opposed to \(14\) - makes a)J 72 759 :M -.095(subtle reference to values of the process )A 263 748 21 15 rC 284 763 :M psb currentpoint pse 263 748 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 292 moveto (t) show 194 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 284 759 :M f56 sf -.109( at two different instants, due to the fact that:)A endp showpage %%Page: 17 17 %%BeginPageSetup initializepage (defalco; pag: 17 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (17)S 204 98 182 34 rC 386 132 :M psb currentpoint pse 204 98 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5824 div 1088 3 -1 roll exch div scale currentpoint translate 64 33 translate -3 607 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (b) show 223 703 moveto 224 ns (j) show 457 607 moveto 384 ns (t) show 572 607 moveto 384 /Times-Roman f1 (,) show 704 607 moveto 384 /Times-Italic f1 (q) show 1050 607 moveto (t) show 907 617 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1178 617 moveto (\)) show 314 635 moveto 384 1000 1364 /Symbol f3 (\() show 1305 635 moveto (\)) show 1518 607 moveto 384 /Symbol f1 (\272) show 1892 607 moveto 384 /Times-Roman f1 (lim) show 1834 846 moveto 224 /Times-Italic f1 (h) show 1965 846 moveto 224 /Symbol f1 (\256) show 2206 846 moveto 224 /Times-Roman f1 (0) show 2343 747 moveto 160 /Symbol f1 (+) show 2521 607 moveto 384 /Times-Italic f1 (E) show 2780 703 moveto 224 ns (t) show 3113 324 moveto 384 ns (q) show 3348 420 moveto 224 ns (j) show 3582 324 moveto 384 ns (t) show 3782 324 moveto 384 /Symbol f1 (+) show 4081 324 moveto 384 /Times-Italic f1 (h) show 3439 334 moveto 384 1000 1171 /Symbol f3 (\() show 4290 334 moveto (\)) show 4488 324 moveto 384 /Symbol f1 (-) show 4780 324 moveto 384 /Times-Italic f1 (q) show 5015 420 moveto 224 ns (j) show 5249 324 moveto 384 ns (t) show 5106 334 moveto 384 1000 1171 /Symbol f3 (\() show 5377 334 moveto (\)) show 4207 900 moveto 384 /Times-Italic f1 (h) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 3090 508 moveto 2425 0 rlineto stroke 2891 355 moveto 384 /Symbol f1 (\346) show 2891 904 moveto (\350) show 2891 685 moveto (\347) show 5527 355 moveto (\366) show 5527 904 moveto (\370) show 5527 685 moveto (\367) show end pse gR gS 17 23 560 797 rC 386 118 :M f56 sf (.)S 72 165 :M -.109(The same is true, incidentally, for the statement)A 72 207 :M ( )S 234 187 122 31 rC 356 218 :M psb currentpoint pse 234 187 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3904 div 992 3 -1 roll exch div scale currentpoint translate 64 -1198 translate -4 1838 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 1934 moveto 224 /Times-Roman f1 (3) show 531 1838 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 1934 moveto 224 /Times-Roman f1 (0) show 388 1868 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 1868 moveto (\)) show 1217 1838 moveto 384 /Symbol f1 (=) show 1621 1838 moveto 384 /Times-Roman f1 (lim) show 1533 2062 moveto 224 /Times-Italic f1 (t) show 1625 2062 moveto 224 /Symbol f1 (-) show 1771 2062 moveto (>) show 1918 2062 moveto (+) show 2065 2062 moveto (\245) show 2312 1572 moveto 384 /Times-Italic f1 (b) show 2517 1668 moveto 224 /Times-Roman f1 (3) show 2800 1572 moveto 384 /Times-Italic f1 (t) show 2915 1572 moveto 384 /Times-Roman f1 (,) show 3047 1572 moveto 384 /Times-Italic f1 (q) show 3393 1572 moveto (t) show 3250 1582 moveto 384 1000 1171 /Symbol f3 (\() show 3521 1582 moveto (\)) show 2657 1600 moveto 384 1000 1364 /Symbol f3 (\() show 3648 1600 moveto (\)) show 2763 2131 moveto 384 /Symbol f2 (e) show 3034 2131 moveto 384 /Symbol f1 (\327) show 3190 2131 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 2283 1739 moveto 1503 0 rlineto stroke end pse gR gS 17 23 560 797 rC 356 207 :M f56 sf ( \(16\))S 72 251 :M .069 .007(suggested by Figure 4.)J 72 287 :M -.089(It is a long-standing problem in stochastic mechanics to understand the role of the assignment it)A 72 304 :M 2.195 .22(attempts of a joint probability law to )J 274 293 21 15 rC 295 308 :M psb currentpoint pse 274 293 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 337 292 moveto (t) show 194 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 465 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 295 304 :M f56 sf 4.031 .403( and )J 328 293 26 15 rC 354 308 :M psb currentpoint pse 328 293 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 359 292 moveto (t) show 487 292 moveto 384 /Times-Roman f1 (\251) show 216 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 600 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 354 304 :M f56 sf 4.172 .417( for )J 384 294 26 12 rC 410 306 :M psb currentpoint pse 384 294 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 832 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate -4 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show 210 261 moveto 384 /Symbol f1 (\271) show 525 261 moveto 384 /Times-Italic f1 (t) show 625 261 moveto 384 /Times-Roman f1 (\251) show end pse gR gS 17 23 560 797 rC 410 304 :M f56 sf 2.489 .249(in terms of transition)J 72 323 :M -.076(probabilities.)A 72 341 :M 1.353 .135(Are such transition probabilities just "a kind of generalised gauge variables necessary to)J 72 359 :M 1.015 .102(express the dynamical content of the theory in the simple and unifying form of stochastic)J 72 377 :M 1.469 .147(variational principles" [12]; or do they, as relation \(15\) seems to suggest, play a role in)J 72 395 :M -.109(modelling the measurement process of non configurational observables ?)A 72 431 :M .119 .012(Here we wish to pursue this point of view a little further, adding the conjecture, that figure 2'')J 72 449 :M -.052(seems to support, that the convergence in \(15\) is almost sure. If this is case, we can write:)A 72 526 :M ( )S 102 483 376 88 rC 478 571 :M psb currentpoint pse 102 483 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12032 div 2816 3 -1 roll exch div scale currentpoint translate 64 38 translate -4 380 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 476 moveto 224 /Times-Roman f1 (3) show 531 380 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 476 moveto 224 /Times-Roman f1 (0) show 388 410 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 410 moveto (\)) show 1217 380 moveto 384 /Symbol f1 (=) show 1621 380 moveto 384 /Times-Roman f1 (lim) show 1533 604 moveto 224 /Times-Italic f1 (t) show 1625 604 moveto 224 /Symbol f1 (-) show 1771 604 moveto (>) show 1918 604 moveto (+) show 2065 604 moveto (\245) show 2274 380 moveto 384 /Times-Italic f1 (q) show 2473 476 moveto 224 /Times-Roman f1 (1) show 2748 380 moveto 384 /Times-Italic f1 (t) show 2605 390 moveto 384 1000 1171 /Symbol f3 (\() show 2876 390 moveto (\)) show 3052 380 moveto 384 /Symbol f1 (\327) show 3209 380 moveto 384 /Times-Italic f1 (b) show 3417 476 moveto 224 /Times-Roman f1 (2) show 3710 380 moveto 384 /Times-Italic f1 (t) show 3825 380 moveto 384 /Times-Roman f1 (,) show 3957 380 moveto 384 /Times-Italic f1 (q) show 4303 380 moveto (t) show 4160 390 moveto 384 1000 1171 /Symbol f3 (\() show 4431 390 moveto (\)) show 3567 408 moveto 384 1000 1364 /Symbol f3 (\() show 4558 408 moveto (\)) show 4756 380 moveto 384 /Symbol f1 (-) show 5048 380 moveto 384 /Times-Italic f1 (q) show 5265 476 moveto 224 /Times-Roman f1 (2) show 5558 380 moveto 384 /Times-Italic f1 (t) show 5415 390 moveto 384 1000 1171 /Symbol f3 (\() show 5686 390 moveto (\)) show 5862 380 moveto 384 /Symbol f1 (\327) show 6019 380 moveto 384 /Times-Italic f1 (b) show 6209 476 moveto 224 /Times-Roman f1 (1) show 6484 380 moveto 384 /Times-Italic f1 (t) show 6599 380 moveto 384 /Times-Roman f1 (,) show 6731 380 moveto 384 /Times-Italic f1 (q) show 7077 380 moveto (t) show 6934 390 moveto 384 1000 1171 /Symbol f3 (\() show 7205 390 moveto (\)) show 6341 408 moveto 384 1000 1364 /Symbol f3 (\() show 7332 408 moveto (\)) show 7531 380 moveto 384 /Symbol f1 (+) show 7825 380 moveto 384 /Symbol f2 (e) show 8096 380 moveto 384 /Symbol f1 (\327) show 8247 380 moveto 384 /Times-Italic f1 (q) show 8461 476 moveto 224 /Times-Roman f1 (3) show 8744 380 moveto 384 /Times-Italic f1 (t) show 8601 390 moveto 384 1000 1171 /Symbol f3 (\() show 8872 390 moveto (\)) show 9048 380 moveto 384 /Symbol f1 (\327) show 9326 380 moveto 384 /Times-Italic f1 (q) show 9525 476 moveto 224 /Times-Roman f1 (1) show 9800 380 moveto 384 /Times-Italic f1 (t) show 9657 390 moveto 384 1000 1171 /Symbol f3 (\() show 9928 390 moveto (\)) show 10060 183 moveto 224 /Times-Roman f1 (2) show 10291 380 moveto 384 /Symbol f1 (+) show 10584 380 moveto 384 /Times-Italic f1 (q) show 10801 476 moveto 224 /Times-Roman f1 (2) show 11094 380 moveto 384 /Times-Italic f1 (t) show 10951 390 moveto 384 1000 1171 /Symbol f3 (\() show 11222 390 moveto (\)) show 11354 183 moveto 224 /Times-Roman f1 (2) show 9188 439 moveto 384 1000 1703 /Symbol f3 (\() show 11513 439 moveto (\)) show 11727 380 moveto 384 /Symbol f1 (=) show -4 1338 moveto (=) show 400 1338 moveto 384 /Times-Roman f1 (lim) show 312 1562 moveto 224 /Times-Italic f1 (t) show 404 1562 moveto 224 /Symbol f1 (-) show 550 1562 moveto (>) show 697 1562 moveto (+) show 844 1562 moveto (\245) show 1052 1099 moveto 384 /Times-Roman f1 (1) show 1093 1631 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1062 1239 moveto 173 0 rlineto stroke 1709 1338 moveto (q) show 1908 1434 moveto 224 /Times-Roman f1 (1) show 2168 1338 moveto 384 /Times-Italic f1 (s) show 2040 1348 moveto 384 1000 1171 /Symbol f3 (\() show 2322 1348 moveto (\)) show 2498 1338 moveto 384 /Symbol f1 (\327) show 2655 1338 moveto 384 /Times-Italic f1 (b) show 2863 1434 moveto 224 /Times-Roman f1 (2) show 3141 1338 moveto 384 /Times-Italic f1 (s) show 3282 1338 moveto 384 /Times-Roman f1 (,) show 3414 1338 moveto 384 /Times-Italic f1 (q) show 3745 1338 moveto (s) show 3617 1348 moveto 384 1000 1171 /Symbol f3 (\() show 3899 1348 moveto (\)) show 3013 1366 moveto 384 1000 1364 /Symbol f3 (\() show 4026 1366 moveto (\)) show 4224 1338 moveto 384 /Symbol f1 (-) show 4516 1338 moveto 384 /Times-Italic f1 (q) show 4733 1434 moveto 224 /Times-Roman f1 (2) show 5011 1338 moveto 384 /Times-Italic f1 (s) show 4883 1348 moveto 384 1000 1171 /Symbol f3 (\() show 5165 1348 moveto (\)) show 5341 1338 moveto 384 /Symbol f1 (\327) show 5498 1338 moveto 384 /Times-Italic f1 (b) show 5688 1434 moveto 224 /Times-Roman f1 (1) show 5948 1338 moveto 384 /Times-Italic f1 (s) show 6089 1338 moveto 384 /Times-Roman f1 (,) show 6221 1338 moveto 384 /Times-Italic f1 (q) show 6552 1338 moveto (s) show 6424 1348 moveto 384 1000 1171 /Symbol f3 (\() show 6706 1348 moveto (\)) show 5820 1366 moveto 384 1000 1364 /Symbol f3 (\() show 6833 1366 moveto (\)) show 7032 1338 moveto 384 /Symbol f1 (+) show 7326 1338 moveto 384 /Symbol f2 (e) show 7597 1338 moveto 384 /Symbol f1 (\327) show 7748 1338 moveto 384 /Times-Italic f1 (q) show 7962 1434 moveto 224 /Times-Roman f1 (3) show 8230 1338 moveto 384 /Times-Italic f1 (s) show 8102 1348 moveto 384 1000 1171 /Symbol f3 (\() show 8384 1348 moveto (\)) show 8560 1338 moveto 384 /Symbol f1 (\327) show 8838 1338 moveto 384 /Times-Italic f1 (q) show 9037 1434 moveto 224 /Times-Roman f1 (1) show 9297 1338 moveto 384 /Times-Italic f1 (s) show 9169 1348 moveto 384 1000 1171 /Symbol f3 (\() show 9451 1348 moveto (\)) show 9583 1141 moveto 224 /Times-Roman f1 (2) show 9814 1338 moveto 384 /Symbol f1 (+) show 10107 1338 moveto 384 /Times-Italic f1 (q) show 10324 1434 moveto 224 /Times-Roman f1 (2) show 10602 1338 moveto 384 /Times-Italic f1 (s) show 10474 1348 moveto 384 1000 1171 /Symbol f3 (\() show 10756 1348 moveto (\)) show 10888 1141 moveto 224 /Times-Roman f1 (2) show 8700 1397 moveto 384 1000 1703 /Symbol f3 (\() show 11047 1397 moveto (\)) show 1572 1436 moveto 384 1000 1979 /Symbol f3 ([) show 11173 1436 moveto (]) show 11290 1338 moveto 384 /Times-Italic f1 (ds) show 1435 1552 moveto 224 /Times-Roman f1 (0) show 1495 1040 moveto 224 /Times-Italic f1 (t) show 1298 1472 moveto 576 /Symbol f1 (\362) show 11723 1338 moveto 384 ns (=) show -4 2398 moveto (=) show 400 2398 moveto 384 /Times-Roman f1 (lim) show 312 2622 moveto 224 /Times-Italic f1 (t) show 404 2622 moveto 224 /Symbol f1 (-) show 550 2622 moveto (>) show 697 2622 moveto (+) show 844 2622 moveto (\245) show 1215 2159 moveto 384 /Times-Roman f1 (1) show 1256 2691 moveto 384 /Times-Italic f1 (t) show 1225 2299 moveto 173 0 rlineto stroke 1759 2398 moveto (q) show 1958 2494 moveto 224 /Times-Roman f1 (1) show 2218 2398 moveto 384 /Times-Italic f1 (s) show 2090 2408 moveto 384 1000 1171 /Symbol f3 (\() show 2372 2408 moveto (\)) show 2548 2398 moveto 384 /Symbol f1 (\327) show 2699 2398 moveto 384 /Times-Italic f1 (dq) show 3108 2494 moveto 224 /Times-Roman f1 (2) show 3386 2398 moveto 384 /Times-Italic f1 (s) show 3258 2408 moveto 384 1000 1171 /Symbol f3 (\() show 3540 2408 moveto (\)) show 3738 2398 moveto 384 /Symbol f1 (-) show 4030 2398 moveto 384 /Times-Italic f1 (q) show 4247 2494 moveto 224 /Times-Roman f1 (2) show 4525 2398 moveto 384 /Times-Italic f1 (s) show 4397 2408 moveto 384 1000 1171 /Symbol f3 (\() show 4679 2408 moveto (\)) show 4855 2398 moveto 384 /Symbol f1 (\327) show 5006 2398 moveto 384 /Times-Italic f1 (dq) show 5397 2494 moveto 224 /Times-Roman f1 (1) show 5657 2398 moveto 384 /Times-Italic f1 (s) show 5529 2408 moveto 384 1000 1171 /Symbol f3 (\() show 5811 2408 moveto (\)) show 1598 2612 moveto 224 /Times-Roman f1 (0) show 1658 2100 moveto 224 /Times-Italic f1 (t) show 1461 2532 moveto 576 /Symbol f1 (\362) show 6010 2398 moveto 384 ns (+) show 1025 2221 moveto (\354) show 1025 2463 moveto (\355) show 1025 2706 moveto (\356) show 6238 2159 moveto 384 /Times-Roman f1 (1) show 6279 2691 moveto 384 /Times-Italic f1 (t) show 6248 2299 moveto 173 0 rlineto stroke 6896 2398 moveto 384 /Symbol f2 (e) show 7167 2398 moveto 384 /Symbol f1 (\327) show 7318 2398 moveto 384 /Times-Italic f1 (q) show 7532 2494 moveto 224 /Times-Roman f1 (3) show 7800 2398 moveto 384 /Times-Italic f1 (s) show 7672 2408 moveto 384 1000 1171 /Symbol f3 (\() show 7954 2408 moveto (\)) show 8130 2398 moveto 384 /Symbol f1 (\327) show 8408 2398 moveto 384 /Times-Italic f1 (q) show 8607 2494 moveto 224 /Times-Roman f1 (1) show 8867 2398 moveto 384 /Times-Italic f1 (s) show 8739 2408 moveto 384 1000 1171 /Symbol f3 (\() show 9021 2408 moveto (\)) show 9153 2201 moveto 224 /Times-Roman f1 (2) show 9384 2398 moveto 384 /Symbol f1 (+) show 9677 2398 moveto 384 /Times-Italic f1 (q) show 9894 2494 moveto 224 /Times-Roman f1 (2) show 10172 2398 moveto 384 /Times-Italic f1 (s) show 10044 2408 moveto 384 1000 1171 /Symbol f3 (\() show 10326 2408 moveto (\)) show 10458 2201 moveto 224 /Times-Roman f1 (2) show 8270 2457 moveto 384 1000 1703 /Symbol f3 (\() show 10617 2457 moveto (\)) show 6758 2496 moveto 384 1000 1979 /Symbol f3 ([) show 10743 2496 moveto (]) show 10860 2398 moveto 384 /Times-Italic f1 (ds) show 6621 2612 moveto 224 /Times-Roman f1 (0) show 6681 2100 moveto 224 /Times-Italic f1 (t) show 6484 2532 moveto 576 /Symbol f1 (\362) show 11211 2221 moveto 384 ns (\374) show 11211 2463 moveto (\375) show 11211 2706 moveto (\376) show end pse gR gS 17 23 560 797 rC 478 526 :M f56 sf ( \(17\))S 72 604 :M -.141(In writing the above relation we have taken into account the fact that)A 95 626 405 60 rC 500 686 :M psb currentpoint pse 95 626 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 12960 div 1920 3 -1 roll exch div scale currentpoint translate 64 45 translate -2 500 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (D) show 386 500 moveto 384 /Times-Italic f1 (t) show 243 510 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 514 510 moveto (\)) show 727 500 moveto 384 /Symbol f1 (\272) show 1035 261 moveto 384 /Times-Roman f1 (1) show 1076 793 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1045 401 moveto 173 0 rlineto stroke 1579 500 moveto (q) show 1778 596 moveto 224 /Times-Roman f1 (1) show 2038 500 moveto 384 /Times-Italic f1 (s) show 1910 510 moveto 384 1000 1171 /Symbol f3 (\() show 2192 510 moveto (\)) show 2368 500 moveto 384 /Symbol f1 (\327) show 2519 500 moveto 384 /Times-Italic f1 (dq) show 2928 596 moveto 224 /Times-Roman f1 (2) show 3206 500 moveto 384 /Times-Italic f1 (s) show 3078 510 moveto 384 1000 1171 /Symbol f3 (\() show 3360 510 moveto (\)) show 3558 500 moveto 384 /Symbol f1 (-) show 3850 500 moveto 384 /Times-Italic f1 (q) show 4067 596 moveto 224 /Times-Roman f1 (2) show 4345 500 moveto 384 /Times-Italic f1 (s) show 4217 510 moveto 384 1000 1171 /Symbol f3 (\() show 4499 510 moveto (\)) show 4675 500 moveto 384 /Symbol f1 (\327) show 4826 500 moveto 384 /Times-Italic f1 (dq) show 5217 596 moveto 224 /Times-Roman f1 (1) show 5477 500 moveto 384 /Times-Italic f1 (s) show 5349 510 moveto 384 1000 1171 /Symbol f3 (\() show 5631 510 moveto (\)) show 1418 714 moveto 224 /Times-Roman f1 (0) show 1478 202 moveto 224 /Times-Italic f1 (t) show 1281 634 moveto 576 /Symbol f1 (\362) show 5829 500 moveto 384 ns (-) show 6120 261 moveto 384 /Times-Roman f1 (1) show 6161 793 moveto 384 /Times-Italic f1 (t) show 6130 401 moveto 173 0 rlineto stroke 6777 500 moveto (q) show 6976 596 moveto 224 /Times-Roman f1 (1) show 7236 500 moveto 384 /Times-Italic f1 (s) show 7108 510 moveto 384 1000 1171 /Symbol f3 (\() show 7390 510 moveto (\)) show 7566 500 moveto 384 /Symbol f1 (\327) show 7723 500 moveto 384 /Times-Italic f1 (b) show 7931 596 moveto 224 /Times-Roman f1 (2) show 8209 500 moveto 384 /Times-Italic f1 (s) show 8350 500 moveto 384 /Times-Roman f1 (,) show 8482 500 moveto 384 /Times-Italic f1 (q) show 8813 500 moveto (s) show 8685 510 moveto 384 1000 1171 /Symbol f3 (\() show 8967 510 moveto (\)) show 8081 528 moveto 384 1000 1364 /Symbol f3 (\() show 9094 528 moveto (\)) show 9292 500 moveto 384 /Symbol f1 (-) show 9584 500 moveto 384 /Times-Italic f1 (q) show 9801 596 moveto 224 /Times-Roman f1 (2) show 10079 500 moveto 384 /Times-Italic f1 (s) show 9951 510 moveto 384 1000 1171 /Symbol f3 (\() show 10233 510 moveto (\)) show 10409 500 moveto 384 /Symbol f1 (\327) show 10566 500 moveto 384 /Times-Italic f1 (b) show 10756 596 moveto 224 /Times-Roman f1 (1) show 11016 500 moveto 384 /Times-Italic f1 (s) show 11157 500 moveto 384 /Times-Roman f1 (,) show 11289 500 moveto 384 /Times-Italic f1 (q) show 11620 500 moveto (s) show 11492 510 moveto 384 1000 1171 /Symbol f3 (\() show 11774 510 moveto (\)) show 10888 528 moveto 384 1000 1364 /Symbol f3 (\() show 11901 528 moveto (\)) show 6640 563 moveto 384 1000 1627 /Symbol f3 ([) show 12027 563 moveto (]) show 6503 714 moveto 224 /Times-Roman f1 (0) show 6563 202 moveto 224 /Times-Italic f1 (t) show 6366 634 moveto 576 /Symbol f1 (\362) show 12121 500 moveto 384 /Times-Roman f1 ( ) show 12217 500 moveto 384 /Times-Italic f1 (ds) show 12650 500 moveto 384 /Symbol f1 (=) show 0 1528 moveto 384 /Times-Roman f1 ( ) show 684 1528 moveto 384 /Symbol f1 (=) show 992 1289 moveto 384 /Times-Roman f1 (1) show 1033 1821 moveto 384 /Times-Italic f1 (t) show 1002 1429 moveto 173 0 rlineto stroke 1536 1528 moveto (q) show 1735 1624 moveto 224 /Times-Roman f1 (1) show 1995 1528 moveto 384 /Times-Italic f1 (s) show 1867 1538 moveto 384 1000 1171 /Symbol f3 (\() show 2149 1538 moveto (\)) show 2325 1528 moveto 384 /Symbol f1 (\327) show 2476 1528 moveto 384 /Times-Italic f1 (dw) show 2960 1624 moveto 224 /Times-Roman f1 (2) show 3238 1528 moveto 384 /Times-Italic f1 (s) show 3110 1538 moveto 384 1000 1171 /Symbol f3 (\() show 3392 1538 moveto (\)) show 3590 1528 moveto 384 /Symbol f1 (-) show 3882 1528 moveto 384 /Times-Italic f1 (q) show 4099 1624 moveto 224 /Times-Roman f1 (2) show 4377 1528 moveto 384 /Times-Italic f1 (s) show 4249 1538 moveto 384 1000 1171 /Symbol f3 (\() show 4531 1538 moveto (\)) show 4707 1528 moveto 384 /Symbol f1 (\327) show 4858 1528 moveto 384 /Times-Italic f1 (dw) show 5324 1624 moveto 224 /Times-Roman f1 (1) show 5584 1528 moveto 384 /Times-Italic f1 (s) show 5456 1538 moveto 384 1000 1171 /Symbol f3 (\() show 5738 1538 moveto (\)) show 1375 1742 moveto 224 /Times-Roman f1 (0) show 1435 1230 moveto 224 /Times-Italic f1 (t) show 1238 1662 moveto 576 /Symbol f1 (\362) show end pse gR gS 17 23 560 797 rC 72 719 :M f56 sf -.132(is a random variable of mean 0 and variance)A 208 741 176 28 rC 384 769 :M psb currentpoint pse 208 741 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5632 div 896 3 -1 roll exch div scale currentpoint translate 64 44 translate -3 500 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (var) show 644 500 moveto 384 /Symbol f1 (D) show 1032 500 moveto 384 /Times-Italic f1 (t) show 889 510 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1160 510 moveto (\)) show 499 528 moveto 384 1000 1364 /Symbol f3 (\() show 1287 528 moveto (\)) show 1501 500 moveto 384 /Symbol f1 (=) show 1888 261 moveto 384 /Times-Roman f1 (1) show 1847 793 moveto 384 /Times-Italic f1 (t) show 1980 622 moveto 224 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1819 401 moveto 331 0 rlineto stroke 2751 500 moveto 384 /Times-Italic f1 (q) show 2950 596 moveto 224 /Times-Roman f1 (1) show 3210 500 moveto 384 /Times-Italic f1 (s) show 3082 510 moveto 384 1000 1171 /Symbol f3 (\() show 3364 510 moveto (\)) show 3496 303 moveto 224 /Times-Roman f1 (2) show 3727 500 moveto 384 /Symbol f1 (+) show 4020 500 moveto 384 /Times-Italic f1 (q) show 4237 596 moveto 224 /Times-Roman f1 (2) show 4515 500 moveto 384 /Times-Italic f1 (s) show 4387 510 moveto 384 1000 1171 /Symbol f3 (\() show 4669 510 moveto (\)) show 4801 303 moveto 224 /Times-Roman f1 (2) show 2613 559 moveto 384 1000 1703 /Symbol f3 (\() show 4960 559 moveto (\)) show 2487 598 moveto 384 1000 1979 /Symbol f3 ([) show 5086 598 moveto (]) show 5203 500 moveto 384 /Times-Italic f1 (ds) show 2350 714 moveto 224 /Times-Roman f1 (0) show 2410 202 moveto 224 /Times-Italic f1 (t) show 2213 634 moveto 576 /Symbol f1 (\362) show end pse gR gS 17 23 560 797 rC 384 758 :M f56 sf (;)S endp showpage %%Page: 18 18 %%BeginPageSetup initializepage (defalco; pag: 18 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (18)S 72 112 :M -.134(we also point out that the harmonic term in)A 205 134 184 28 rC 389 162 :M psb currentpoint pse 205 134 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5888 div 896 3 -1 roll exch div scale currentpoint translate 64 -1228 translate -3 1772 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 193 1868 moveto 224 /Times-Roman f1 (1) show 299 1868 moveto 224 /Times-Roman f1 (,) show 370 1868 moveto 224 /Times-Roman f1 (2) show 616 1772 moveto 384 /Symbol f1 (=) show 955 1533 moveto 384 /Times-Roman f1 (1) show 955 2065 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 934 1673 moveto 235 0 rlineto stroke 1233 1772 moveto 384 /Times-Italic f1 (p) show 1421 1868 moveto 224 /Times-Roman f1 (1) show 1535 1606 moveto (2) show 1766 1772 moveto 384 /Symbol f1 (+) show 2089 1533 moveto 384 /Times-Roman f1 (1) show 2089 2065 moveto (2) show 2068 1673 moveto 235 0 rlineto stroke 2367 1772 moveto 384 /Times-Italic f1 (p) show 2573 1868 moveto 224 /Times-Roman f1 (2) show 2705 1606 moveto (2) show 2936 1772 moveto 384 /Symbol f1 (+) show 3259 1533 moveto 384 /Times-Roman f1 (1) show 3259 2065 moveto (2) show 3238 1673 moveto 235 0 rlineto stroke 3521 1772 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (w) show 3847 1601 moveto 224 /Times-Roman f1 (2) show 4055 1772 moveto 384 /Symbol f1 (\327) show 4333 1772 moveto 384 /Times-Italic f1 (q) show 4532 1868 moveto 224 /Times-Roman f1 (1) show 4646 1606 moveto (2) show 4877 1772 moveto 384 /Symbol f1 (+) show 5170 1772 moveto 384 /Times-Italic f1 (q) show 5387 1868 moveto 224 /Times-Roman f1 (2) show 5519 1606 moveto (2) show 4195 1814 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1515 /Symbol f3 (\() show 5678 1814 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 195 :M f56 sf .257 .026(has been introduced into the model just in order to prevent spreading of the wave function in)J 72 212 :M -.22(the )A 89 202 27 14 rC 116 216 :M psb currentpoint pse 89 202 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 357 moveto 224 /Times-Roman f1 (1) show 313 261 moveto 384 /Times-Roman f1 (,) show 448 261 moveto 384 /Times-Italic f1 (x) show 657 357 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 116 212 :M f56 sf ( plane.)S 72 247 :M -.045(The intuition that \(17\) provides on the time progress of the process of measurement of )A 485 235 38 18 rC 523 253 :M psb currentpoint pse 485 235 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 576 3 -1 roll exch div scale currentpoint translate 64 -206 translate -4 590 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 686 moveto 224 /Times-Roman f1 (3) show 531 590 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 844 686 moveto 224 /Times-Roman f1 (0) show 388 620 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1388 /Symbol f3 (\() show 1003 620 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 268 :M f56 sf -.037(is shown by Figure 5 and 6. We recall, for comparison, that for the analogous classical system,)A 72 286 :M -.134(canonical angular momentum is a linear combination of area per unit time spanned by the vector)A 72 290 61 17 rC 133 307 :M psb currentpoint pse 72 290 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1952 div 544 3 -1 roll exch div scale currentpoint translate 64 59 translate 118 325 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 317 421 moveto 224 /Times-Roman f1 (1) show 592 325 moveto 384 /Times-Italic f1 (t) show 449 335 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 720 335 moveto (\)) show 834 325 moveto 384 /Times-Roman f1 (,) show 966 325 moveto 384 /Times-Italic f1 (q) show 1183 421 moveto 224 /Times-Roman f1 (2) show 1476 325 moveto 384 /Times-Italic f1 (t) show 1333 335 moveto 384 1000 1171 /Symbol f3 (\() show 1604 335 moveto (\)) show -20 353 moveto 384 1000 1369 /Symbol f3 (\() show 1731 353 moveto (\)) show end pse gR gS 17 23 560 797 rC 133 302 :M f56 sf -.13( and magnetic flux linked to the trajectory.)A 175 341 :M .674(a.)A 92 380 176 108 rC 176 108 8 528 [176 0 0 108 92 380] F T 1 F :f :l FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000000000000000FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000FFFFFF000000FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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:j gR gS 17 23 560 797 rC 1 65.5 525 @w 1 65.5 525 @w 225 66 525.5 @c 1 291.5 525 @w 226 292 525.5 @c 1 518.5 525 @w 1 518.5 525 @w 165 65.5 526 @w 1 65.5 691 @w 1 65.5 691 @w 225 66 691.5 @c 165 291.5 526 @w 1 291.5 691 @w 226 292 691.5 @c 165 518.5 526 @w 1 518.5 691 @w 1 518.5 691 @w 277 707 :M f56 sf -.046(Figure 6)A 74 729 :M f193 sf -.17(The behaviour of the pointer variable )A 251 711 38 31 rC 289 742 :M psb currentpoint pse 251 711 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1216 div 992 3 -1 roll exch div scale currentpoint translate 64 45 translate 158 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 394 388 moveto 224 /Times-Roman f1 (3) show 677 292 moveto 384 /Times-Italic f1 (t) show 534 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 833 302 moveto (\)) show 24 824 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 295 824 moveto 384 /Symbol f1 (\327) show 451 824 moveto 384 /Times-Italic f1 (t) show 612 653 moveto 224 /Times-Roman f1 (2) show 893 824 moveto 384 ns (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -130 406 886 501 vb 16 th 0 432 moveto 1107 0 rlineto stroke end pse gR gS 17 23 560 797 rC 289 729 :M f193 sf -.078( as a function of )A 368 721 7 10 rC 375 731 :M psb currentpoint pse 368 721 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -606 -36 translate 666 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 375 729 :M f193 sf -.022(shown in )A f56 sf (a.)S f193 sf -.023( corresponds to the)A 192 755 :M -.211(sample path )A 251 743 88 18 rC 339 761 :M psb currentpoint pse 251 743 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2816 div 576 3 -1 roll exch div scale currentpoint translate 64 56 translate 118 328 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 317 424 moveto 224 /Times-Roman f1 (1) show 592 328 moveto 384 /Times-Italic f1 (t) show 449 338 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 720 338 moveto (\)) show 834 328 moveto 384 /Times-Roman f1 (,) show 966 328 moveto 384 /Times-Italic f1 (q) show 1183 424 moveto 224 /Times-Roman f1 (2) show 1476 328 moveto 384 /Times-Italic f1 (t) show 1333 338 moveto 384 1000 1171 /Symbol f3 (\() show 1604 338 moveto (\)) show 1718 328 moveto 384 /Times-Roman f1 (,) show 1850 328 moveto 384 /Times-Italic f1 (q) show 2064 424 moveto 224 /Times-Roman f1 (3) show 2347 328 moveto 384 /Times-Italic f1 (t) show 2204 338 moveto 384 1000 1171 /Symbol f3 (\() show 2475 338 moveto (\)) show -20 358 moveto 384 1000 1388 /Symbol f3 (\() show 2602 358 moveto (\)) show end pse gR gS 17 23 560 797 rC 339 755 :M f193 sf .602 .06( shown in )J f56 sf .221(b.)A f193 sf (.)S endp showpage %%Page: 19 19 %%BeginPageSetup initializepage (defalco; pag: 19 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (19)S 72 112 :M -.048(As to the "physical reality" of trajectories such as those shown in Figures 5 and 6 , we refer the)A 72 130 :M -.088(reader to the cautionary remarks in the beautifully lucid introduction of Reference [13]. We just)A 72 148 :M -.088(wish to point out that, as it is clear from the analysis of the measurement process given by Pauli)A 72 166 :M .271 .027([14], the case of scattering by a scalar potential )J 305 157 12 11 rC 317 168 :M psb currentpoint pse 305 157 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 384 div 352 3 -1 roll exch div scale currentpoint translate 64 27 translate -7 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (V) show end pse gR gS 17 23 560 797 rC 317 166 :M f56 sf .446 .045( does not exhaust the instances in which a)J 72 184 :M -.082(particle description of quantum behaviour is asymptotically appropriate; the case of motion in a)A 72 202 :M -.186(vector potential )A 147 192 12 11 rC 159 203 :M psb currentpoint pse 147 192 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 384 div 352 3 -1 roll exch div scale currentpoint translate 64 -625 translate -5 945 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (A) show end pse gR gS 17 23 560 797 rC 159 202 :M f56 sf .223 .022(deserves a similar attention. Also in this case, therefore, a comparison with)J 72 220 :M -.1(the paths-of-physical-particles picture of stochastic mechanics is in order.)A 72 238 :M -.001(Reference [13] makes it clear \(through the sharp statement that, for those scattering diffusions)A 72 242 21 15 rC 93 257 :M psb currentpoint pse 72 242 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate -1286 60 translate 1341 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 1687 292 moveto (t) show 1544 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1815 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 93 253 :M f56 sf -.189( that admit a final linear momentum )A 262 242 73 20 rC 335 262 :M psb currentpoint pse 262 242 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2336 div 640 3 -1 roll exch div scale currentpoint translate 64 60 translate 0 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 210 388 moveto 224 /Symbol f1 (+) show 470 292 moveto 384 ns (=) show 781 292 moveto 384 /Times-Roman f1 (lim) show 797 516 moveto 224 /Times-Italic f1 (t) show 881 516 moveto 224 /Symbol f1 (\256) show 1122 516 moveto (\245) show 1341 292 moveto 384 /Times-Italic f1 (q) show 1687 292 moveto (t) show 1544 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1815 302 moveto (\)) show 2107 292 moveto 384 /Times-Italic f1 (t) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 387 2075 0 vb end pse gR gS 17 23 560 797 rC 335 253 :M f56 sf .165 .016(, any random variable measurable with)J 72 276 :M -.055(respect to the tail )A f225 sf -.089(s-)A f56 sf -.061(algebra is a function of )A 282 266 14 14 rC 296 280 :M psb currentpoint pse 282 266 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 448 3 -1 roll exch div scale currentpoint translate 64 28 translate 0 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 210 388 moveto 224 /Symbol f1 (+) show end pse gR gS 17 23 560 797 rC 296 276 :M f56 sf .202 .02(\) that no contradiction emerges in the case of a)J 72 295 :M -.123(scalar potential.)A 72 313 :M .651 .065(If equalities \(15\), \(16\), \(17\) can, as equality \(14\), be proven to be true beyond the heuristic)J 72 331 :M 1.13 .113(level at which we have introduced them, the question arises of characterising the random)J 72 348 :M -.075(variables which are measurable with respect to the tail )A f225 sf -.11(s-)A f56 sf -.079(algebra generated by the process )A 502 337 21 15 rC 523 352 :M psb currentpoint pse 502 337 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 480 3 -1 roll exch div scale currentpoint translate -1286 60 translate 1341 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 1687 292 moveto (t) show 1544 302 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 1815 302 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 367 :M f56 sf -.049(studied in Section 3.)A 72 385 :M .007 .001(Stated in physical terms this question can be reformulated as: What else, beyond a component)J 72 403 :M -.082(of angular momentum, can be "measured" through the long-time behaviour of a Nelson process)A 72 421 :M .257 .026(in an inhomogeous magnetic field ? Making the obvious answer \(energy of the motion in the)J 72 428 27 14 rC 99 442 :M psb currentpoint pse 72 428 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 -13382 translate -6 13702 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 13798 moveto 224 /Times-Roman f1 (1) show 313 13702 moveto 384 /Times-Roman f1 (,) show 448 13702 moveto 384 /Times-Italic f1 (x) show 657 13798 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 99 438 :M f56 sf -.038( plane, as suggested by the quantum mechanical statement of simultaneous measurability)A 72 455 :M 1.672 .167(of )J 87 445 18 15 rC 105 460 :M psb currentpoint pse 87 445 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 576 div 480 3 -1 roll exch div scale currentpoint translate 64 -1374 translate -3 1694 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (h) show 193 1790 moveto 224 /Times-Roman f1 (1) show 299 1790 moveto 224 /Times-Roman f1 (,) show 370 1790 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 105 455 :M f56 sf 2.364 .236( and )J 134 445 14 15 rC 148 460 :M psb currentpoint pse 134 445 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 448 div 480 3 -1 roll exch div scale currentpoint translate 64 -4057 translate -4 4377 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 248 4473 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 148 455 :M f56 sf 1.353 .135(\) precise requires further research. We are presently studying a stochastic)J 72 476 :M -.164(analogue of the classical adiabatic invariant )A 280 461 91 24 rC 371 485 :M psb currentpoint pse 280 461 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2912 div 768 3 -1 roll exch div scale currentpoint translate 64 -1111 translate 249 1591 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (p) show 437 1687 moveto 224 /Times-Roman f1 (1) show 627 1591 moveto 384 /Symbol f1 (\327) show 778 1591 moveto 384 /Times-Italic f1 (dq) show 1169 1687 moveto 224 /Times-Roman f1 (1) show 1382 1591 moveto 384 /Symbol f1 (+) show 1684 1591 moveto 384 /Times-Italic f1 (p) show 1890 1687 moveto 224 /Times-Roman f1 (2) show 2098 1591 moveto 384 /Symbol f1 (\327) show 2249 1591 moveto 384 /Times-Italic f1 (dq) show 2658 1687 moveto 224 /Times-Roman f1 (2) show -1 1751 moveto 640 /Symbol f1 (\362) show /ellipse { setlinewidth newpath matrix currentmatrix 5 1 roll translate scale 0 0 1 0 360 arc setmatrix stroke } def 84 84 92 1492 16 ellipse end pse gR gS 17 23 560 797 rC 371 476 :M f56 sf -.052(, carrying energy information in)A 72 498 :M 1.149 .115(much the same way as )J 191 487 28 16 rC 219 503 :M psb currentpoint pse 191 487 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 896 div 512 3 -1 roll exch div scale currentpoint translate 64 -1313 translate -2 1665 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Symbol f1 (L) show 282 1761 moveto 224 /Times-Roman f1 (3) show 565 1665 moveto 384 /Times-Italic f1 (t) show 422 1675 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 693 1675 moveto (\)) show end pse gR gS 17 23 560 797 rC 219 498 :M f56 sf 1.095 .109( carries angular momentum information. In carrying out this)J 72 518 :M -.118(program it is not any more sufficient to consider the Hamiltonian)A 72 557 :M ( )S 129 540 345 28 rC 474 568 :M psb currentpoint pse 129 540 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 11040 div 896 3 -1 roll exch div scale currentpoint translate 64 44 translate -7 500 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 285 596 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (w) show 478 596 moveto 224 /Times-Roman f1 (,) show 551 596 moveto 224 /Symbol f2 (e) show 808 500 moveto 384 /Symbol f1 (=) show 1147 261 moveto 384 /Times-Roman f1 (1) show 1147 793 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1126 401 moveto 235 0 rlineto stroke 1425 500 moveto 384 /Times-Italic f1 (p) show 1613 596 moveto 224 /Times-Roman f1 (1) show 1727 334 moveto (2) show 1958 500 moveto 384 /Symbol f1 (+) show 2281 261 moveto 384 /Times-Roman f1 (1) show 2281 793 moveto (2) show 2260 401 moveto 235 0 rlineto stroke 2559 500 moveto 384 /Times-Italic f1 (p) show 2765 596 moveto 224 /Times-Roman f1 (2) show 2897 334 moveto (2) show 3128 500 moveto 384 /Symbol f1 (+) show 3451 261 moveto 384 /Times-Roman f1 (1) show 3451 793 moveto (2) show 3430 401 moveto 235 0 rlineto stroke 3713 500 moveto 384 /Symbol f2 (w) show 4039 329 moveto 224 /Times-Roman f1 (2) show 4247 500 moveto 384 /Symbol f1 (\327) show 4525 500 moveto 384 /Times-Italic f1 (q) show 4724 596 moveto 224 /Times-Roman f1 (1) show 4838 334 moveto (2) show 5069 500 moveto 384 /Symbol f1 (+) show 5362 500 moveto 384 /Times-Italic f1 (q) show 5579 596 moveto 224 /Times-Roman f1 (2) show 5711 334 moveto (2) show 4387 542 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1515 /Symbol f3 (\() show 5870 542 moveto (\)) show 6069 500 moveto 384 /Symbol f1 (+) show 6392 261 moveto 384 /Times-Roman f1 (1) show 6392 793 moveto (2) show 6371 401 moveto 235 0 rlineto stroke 6670 500 moveto 384 /Times-Italic f1 (p) show 6873 596 moveto 224 /Times-Roman f1 (3) show 6995 334 moveto (2) show 7225 500 moveto 384 /Symbol f1 (-) show 7518 500 moveto 384 /Symbol f2 (e) show 7789 500 moveto 384 /Symbol f1 (\327) show 7940 500 moveto 384 /Times-Italic f1 (q) show 8154 596 moveto 224 /Times-Roman f1 (3) show 8352 500 moveto 384 /Symbol f1 (\327) show 8630 500 moveto 384 /Times-Italic f1 (q) show 8829 596 moveto 224 /Times-Roman f1 (1) show 9019 500 moveto 384 /Symbol f1 (\327) show 9179 500 moveto 384 /Times-Italic f1 (p) show 9385 596 moveto 224 /Times-Roman f1 (2) show 9615 500 moveto 384 /Symbol f1 (-) show 9907 500 moveto 384 /Times-Italic f1 (q) show 10124 596 moveto 224 /Times-Roman f1 (2) show 10332 500 moveto 384 /Symbol f1 (\327) show 10492 500 moveto 384 /Times-Italic f1 (p) show 10680 596 moveto 224 /Times-Roman f1 (1) show 8492 528 moveto 384 1000 1369 /Symbol f3 (\() show 10821 528 moveto (\)) show end pse gR gS 17 23 560 797 rC 474 557 :M f56 sf .164 .016( \(6'\))J 72 582 :M .046 .005(adopted up to this point. In describing the coupling of the harmonic "system" )J 446 572 34 14 rC 480 586 :M psb currentpoint pse 446 572 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 448 3 -1 roll exch div scale currentpoint translate 64 -966 translate -9 1286 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 190 1382 moveto 224 /Times-Roman f1 (1) show 310 1286 moveto 384 /Times-Roman f1 ( ) show 414 1286 moveto (,) show 555 1286 moveto ( ) show 651 1286 moveto 384 /Times-Italic f1 (q) show 868 1382 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 480 582 :M f56 sf .36 .036( with the)J 72 599 :M -.069("apparatus" )A 130 589 13 15 rC 143 604 :M psb currentpoint pse 130 589 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 480 3 -1 roll exch div scale currentpoint translate -596 -966 translate 651 1286 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 865 1382 moveto 224 /Times-Roman f1 (3) show end pse gR gS 17 23 560 797 rC 143 599 :M f56 sf -.141( through the vector potential)A 248 605 98 54 rC 346 659 :M psb currentpoint pse 248 605 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3136 div 1728 3 -1 roll exch div scale currentpoint translate 64 -6023 translate 190 6407 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (A) show 469 6503 moveto 224 /Times-Roman f1 (1) show 742 6407 moveto 384 /Times-Italic f1 (x) show 601 6417 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 946 6417 moveto (\)) show 1160 6407 moveto 384 /Symbol f1 (=) show 1474 6407 moveto (-) show 1703 6407 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (e) show 1974 6407 moveto 384 /Symbol f1 (\327) show 2128 6407 moveto 384 /Times-Italic f1 (x) show 2337 6503 moveto 224 /Times-Roman f1 (2) show 2545 6407 moveto 384 /Symbol f1 (\327) show 2699 6407 moveto 384 /Times-Italic f1 (x) show 2905 6503 moveto 224 /Times-Roman f1 (3) show 259 6983 moveto 384 /Times-Italic f1 (A) show 556 7079 moveto 224 /Times-Roman f1 (2) show 847 6983 moveto 384 /Times-Italic f1 (x) show 706 6993 moveto 384 1000 1171 /Symbol f3 (\() show 1051 6993 moveto (\)) show 1265 6983 moveto 384 /Symbol f1 (=) show 1575 6983 moveto 384 /Symbol f2 (e) show 1846 6983 moveto 384 /Symbol f1 (\327) show 2000 6983 moveto 384 /Times-Italic f1 (x) show 2191 7079 moveto 224 /Times-Roman f1 (1) show 2381 6983 moveto 384 /Symbol f1 (\327) show 2535 6983 moveto 384 /Times-Italic f1 (x) show 2741 7079 moveto 224 /Times-Roman f1 (3) show 2869 6983 moveto 384 /Times-Roman f1 ( ) show 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f1 (k) show 220 3293 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (e) show 365 3194 moveto 160 /Times-Roman f1 (2) show 613 3142 moveto 384 /Symbol f1 (=) show 952 2903 moveto 384 /Times-Roman f1 (1) show 952 3435 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 931 3043 moveto 235 0 rlineto stroke 1236 3142 moveto 384 /Symbol f1 (\327) show 1388 3142 moveto 384 /Symbol f2 (e) show 1615 2971 moveto 224 /Times-Roman f1 (2) show 1823 3142 moveto 384 /Symbol f1 (\327) show 1974 3142 moveto 384 /Times-Italic f1 (q) show 2188 3238 moveto 224 /Times-Roman f1 (3) show 2310 2976 moveto (2) show 2518 3142 moveto 384 /Symbol f1 (\327) show 2796 3142 moveto 384 /Times-Italic f1 (q) show 2995 3238 moveto 224 /Times-Roman f1 (1) show 3109 2976 moveto (2) show 3340 3142 moveto 384 /Symbol f1 (+) show 3633 3142 moveto 384 /Times-Italic f1 (q) show 3850 3238 moveto 224 /Times-Roman f1 (2) show 3982 2976 moveto (2) show 2658 3184 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1515 /Symbol f3 (\() show 4141 3184 moveto (\)) show end pse gR gS 17 23 560 797 rC 72 721 :M f56 sf -.047(responsible of transfer of energy between "system" and "apparatus".)A endp showpage %%Page: 20 20 %%BeginPageSetup initializepage (defalco; pag: 20 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (20)S 72 94 :M -.005(Figure 7 gives \(in a simple adiabatic approximation\) a hint of the new phenomena \(relevant to)A 72 112 :M .807 .081(the problem of understanding the meaning itself of "quantization" and "superposition" in a)J 72 127 :M .717 .072(stochastic context\) that appear in stochastic evolution under )J 372 117 52 16 rC 424 133 :M psb currentpoint pse 372 117 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1664 div 512 3 -1 roll exch div scale currentpoint translate 64 -2583 translate -7 2903 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 285 2999 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (w) show 478 2999 moveto 224 /Times-Roman f1 (,) show 551 2999 moveto 224 /Symbol f2 (e) show 793 2903 moveto 384 /Symbol f1 (+) show 1095 2903 moveto 384 /Times-Italic f1 (k) show 1315 3054 moveto 224 /Symbol f2 (e) show 1460 2955 moveto 160 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 424 127 :M f56 sf 1.423 .142(: the dominant new)J 72 147 :M -.017(feature is now that the motion in the )A 248 137 31 14 rC 279 151 :M psb currentpoint pse 248 137 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 992 div 448 3 -1 roll exch div scale currentpoint translate 64 59 translate -6 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (x) show 185 357 moveto 224 /Times-Roman f1 (1) show 313 261 moveto 384 /Times-Roman f1 (,) show 454 261 moveto ( ) show 550 261 moveto 384 /Times-Italic f1 (x) show 759 357 moveto 224 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 279 147 :M f56 sf .406 .041( plane visits in turn, at random times, the possible)J 72 164 :M .18 .018("quantized" values of )J 179 152 165 17 rC 344 169 :M psb currentpoint pse 179 152 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5280 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 324 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 190 420 moveto 224 /Times-Roman f1 (1) show 465 324 moveto 384 /Times-Italic f1 (t) show 322 334 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 593 334 moveto (\)) show 769 324 moveto 384 /Symbol f1 (\327) show 929 324 moveto 384 /Times-Italic f1 (p) show 1135 420 moveto 224 /Times-Roman f1 (2) show 1428 324 moveto 384 /Times-Italic f1 (t) show 1543 324 moveto 384 /Times-Roman f1 (,) show 1675 324 moveto 384 /Times-Italic f1 (q) show 2021 324 moveto (t) show 1878 334 moveto 384 1000 1171 /Symbol f3 (\() show 2149 334 moveto (\)) show 1285 352 moveto 384 1000 1364 /Symbol f3 (\() show 2276 352 moveto (\)) show 2474 324 moveto 384 /Symbol f1 (-) show 2766 324 moveto 384 /Times-Italic f1 (q) show 2983 420 moveto 224 /Times-Roman f1 (2) show 3276 324 moveto 384 /Times-Italic f1 (t) show 3133 334 moveto 384 1000 1171 /Symbol f3 (\() show 3404 334 moveto (\)) show 3580 324 moveto 384 /Symbol f1 (\327) show 3740 324 moveto 384 /Times-Italic f1 (p) show 3928 420 moveto 224 /Times-Roman f1 (1) show 4203 324 moveto 384 /Times-Italic f1 (t) show 4318 324 moveto 384 /Times-Roman f1 (,) show 4450 324 moveto 384 /Times-Italic f1 (q) show 4796 324 moveto (t) show 4653 334 moveto 384 1000 1171 /Symbol f3 (\() show 4924 334 moveto (\)) show 4060 352 moveto 384 1000 1364 /Symbol f3 (\() show 5051 352 moveto (\)) show end pse gR gS 17 23 560 797 rC 344 164 :M f56 sf .332 .033( present in the initial "superposition")J 72 184 :M -.106(of states, alternating these visits with transient episodes similar to the ones depicted in the initial)A 72 202 :M (part of Figure 2.)S 152 224 282 174 rC 282 174 8 846 [282 0 0 174 152 224] F T 1 F :f :l FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF 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3 1 roll sub 5280 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate -9 324 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (q) show 190 420 moveto 224 /Times-Roman f1 (1) show 465 324 moveto 384 /Times-Italic f1 (t) show 322 334 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() show 593 334 moveto (\)) show 769 324 moveto 384 /Symbol f1 (\327) show 929 324 moveto 384 /Times-Italic f1 (p) show 1135 420 moveto 224 /Times-Roman f1 (2) show 1428 324 moveto 384 /Times-Italic f1 (t) show 1543 324 moveto 384 /Times-Roman f1 (,) show 1675 324 moveto 384 /Times-Italic f1 (q) show 2021 324 moveto (t) show 1878 334 moveto 384 1000 1171 /Symbol f3 (\() show 2149 334 moveto (\)) show 1285 352 moveto 384 1000 1364 /Symbol f3 (\() show 2276 352 moveto (\)) show 2474 324 moveto 384 /Symbol f1 (-) show 2766 324 moveto 384 /Times-Italic f1 (q) show 2983 420 moveto 224 /Times-Roman f1 (2) show 3276 324 moveto 384 /Times-Italic f1 (t) show 3133 334 moveto 384 1000 1171 /Symbol f3 (\() show 3404 334 moveto (\)) show 3580 324 moveto 384 /Symbol f1 (\327) show 3740 324 moveto 384 /Times-Italic f1 (p) show 3928 420 moveto 224 /Times-Roman f1 (1) show 4203 324 moveto 384 /Times-Italic f1 (t) show 4318 324 moveto 384 /Times-Roman f1 (,) show 4450 324 moveto 384 /Times-Italic f1 (q) show 4796 324 moveto (t) show 4653 334 moveto 384 1000 1171 /Symbol f3 (\() show 4924 334 moveto (\)) show 4060 352 moveto 384 1000 1364 /Symbol f3 (\() show 5051 352 moveto (\)) show end pse gR gS 17 23 560 797 rC 261 424 :M f193 sf -.078( as a function of )A 340 416 7 10 rC 347 426 :M psb currentpoint pse 340 416 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 224 div 320 3 -1 roll exch div scale currentpoint translate -606 -36 translate 666 292 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (t) show end pse gR gS 17 23 560 797 rC 347 424 :M f193 sf -.082( for evolution under )A 444 414 52 16 rC 496 430 :M psb currentpoint pse 444 414 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1664 div 512 3 -1 roll exch div scale currentpoint translate 64 -2583 translate -7 2903 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (H) show 285 2999 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 224 /Symbol f2 (w) show 478 2999 moveto 224 /Times-Roman f1 (,) show 551 2999 moveto 224 /Symbol f2 (e) show 793 2903 moveto 384 /Symbol f1 (+) show 1095 2903 moveto 384 /Times-Italic f1 (k) show 1315 3054 moveto 224 /Symbol f2 (e) show 1460 2955 moveto 160 /Times-Roman f1 (2) show end pse gR gS 17 23 560 797 rC 85 443 :M f193 sf -.057(from the initial condition given by a superposition, with equal weights, of )A 436 433 74 15 rC 510 448 :M psb currentpoint pse 436 433 :M psb 30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2368 div 480 3 -1 roll exch div scale currentpoint translate 64 59 translate -13 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (j) show 256 357 moveto 224 /Times-Roman f1 (0) show 380 357 moveto 224 /Times-Roman f1 (,) show 452 357 moveto 224 /Times-Roman f1 (0) show 598 261 moveto 384 /Times-Roman f1 (,) show 739 261 moveto ( ) show 835 261 moveto 384 /Symbol f2 (j) show 1103 357 moveto 224 /Times-Roman f1 (2) show 1227 357 moveto 224 /Times-Roman f1 (,) show 1299 357 moveto 224 /Times-Roman f1 (0) show 1445 261 moveto 384 /Times-Roman f1 (,) show 1586 261 moveto ( ) show 1682 261 moveto 384 /Symbol f2 (j) show 1954 357 moveto 224 /Times-Roman f1 (4) show 2077 357 moveto 224 /Times-Roman f1 (,) show 2149 357 moveto 224 /Times-Roman f1 (0) show end pse gR gS 17 23 560 797 rC 72 481 :M f67 sf -.079(Acknowledgements)A 72 499 :M f56 sf .812 .081(It is a pleasure to thank Ludwig Streit and Daniele Mundici for decisive encouragement at)J 72 517 :M -.071(different stages of this project.)A endp showpage %%Page: 21 21 %%BeginPageSetup initializepage (defalco; pag: 21 di 21)setjob %%EndPageSetup -17 -23 :T gS 17 23 560 797 rC 291 802 :M f56 sf (21)S 70 97 :M ( )S f67 sf -.018(References)A 70 124 :M f56 sf -.041([1] E. Nelson: "Dynamical Theories of Brownian Motion" Princeton University Press, 1967.)A 70 148 :M -.039([2] E. Nelson: Bulletin of the American Mathematical Society, )A f103 sf -.048(84)A f56 sf -.042(, 121-124, 1970.)A 70 172 :M -.019([3] E. Nelson: "Quantum Fluctuations" Princeton University Press, 1985.)A 70 196 :M .273 .027([4] W. Gerlach, O. Stern: Annalen der Physik. )J f103 sf .135(LXXIV)A f56 sf .395 .039(, 673-699, 1924.)J 70 220 :M .383 .038([5] W. Faris: Foundations of Physics, )J f103 sf .146(12)A f56 sf .465 .046(, 1-26, 1982.)J 70 244 :M -.019([6] R. Feynman, A. Hibbs: "Quantum Mechanics and Path Integrals", McGraw-Hill, 1965.)A 70 268 :M -.003([7] D. de Falco: Letters in Mathematical Physics, )A f103 sf (14)S f56 sf (, 41-45, 1987.)S 70 292 :M .159 .016([8] D. Shucker: Journal of Functional Analysis, )J f103 sf .056(38)A f56 sf .212 .021(, 146-155, 1980.)J 70 316 :M .14 .014([9] E. Carlen: Annales de l'Institut Henri Poincar\216, )J f103 sf .052(42)A f56 sf .194 .019(, 407-428, 1985.)J 70 340 :M .989 .099([10] A. Posilicano, S. Ugolini: "The Flux-across-surfaces Theorem: a Probabilistic Proof",)J 70 352 :M .272 .027(Preprint, Universit\210 dell'Insubria, 1999.)J 70 376 :M .373 .037([11] F. Guerra: Journal de Physique, )J f103 sf .122(34)A f56 sf .363 .036(, C1-95 - C1-98, 1973.)J 70 400 :M -.026([12] F. Guerra: "The Problem of the Physical Interpretation of Nelson Stochastic Mechanics as)A 70 412 :M -.069(a Model for Quantum Mechanics", Preprint, Universit\210 di Roma "La Sapienza", 1997.)A 70 436 :M .286 .029([13] E. Carlen: Lecture Notes in Physics, )J f103 sf .096(262)A f56 sf .361 .036(, 139-147, 1986.)J 70 460 :M .782 .078([14] W. Pauli: "General Principles of Quantum Mechanics" \(English translation\), Springer-)J 70 472 :M .163 .016(Verlag, 1980.)J endp showpage %%Trailer end %%EOF ---------------0003081043908--