Towrds Foundtionl Principl for Quntum Mchnics Th wind is not moving, th flg is not moving. Mind is moving. Trnc J. Nlson A simpl foundtionl principl for quntum mchnics is proposd lding nturlly to compl probbility dnsity prssd s sum of pth intgrls. Th ction is thn drivd in its rltivisticlly-invrint form with th Lgrngin following from th principl instd of bing imposd s n tr hypothsis. Howvr, th sum is rstrictd to physicl pths on which th prticl hs rl vctor momntum. This rstriction vidntly conflicts with th usul drivtion of Schrodingr's qution nd from it consrvtion of nrgy nd probbility s dfind by Born. To s if th rstrictd pth intgrl sum lds to snsibl prdictions, th doubl-slit primnt is nlyzd using singl-quntum trnsfr pproimtion. Th rsult is found to b in gnrl grmnt with primnt providd w dd nturl msurmnt hypothsis. Howvr, it dos prdict tht nrgy is not ctly consrvd. Th most probbl chng in nrgy (nr th position of th first minimum t th scrn is loss tht is proportionl to th squr of Plnck's constnt whn prssd in trms of th slit gomtry nd th mss of th prticl. Introduction Whn I first hrd bout Hisnbrg's uncrtinty principl, I nivly supposd tht th sttisticl ntur of th prdictions of quntum mchnics would mrg from th unprdictbility of individul chngs of qunt btwn filds nd mtril prticls. Howvr, th dvlopmnt didn't tk this pth. Wht nsud ws bst chrctrizd by Richrd Fynmn whn h sid [] "w cnnot mk th mystry go wy w will just tll you how it works." Aftr thinking bout it for long tim, I my hv found simpl wy forwrd long th pth I hd originlly nticiptd. Th first prt of th foundtionl principl (notbly missing from convntionl quntum thory [] tht I wnt to propos is: Prticls nd filds intrct discrtly through quntizd chngs of momntum tht occur t prdictbl rts but t unprdictbl tims. I should dd tht I think this is obvious bcus on could only discovr th timing of first chng with dditionl intrctions tht would compound th sitution.. Building on th Foundtion Th forc cting on prticl is th ngtiv of th grdint of th potntil nrgy. Whn th potntil, sy V(, is prssd s Fourir intgrl, th intgrnd diffrs from tht of th potntil by fctor of ik, Zn Flsh, Zn Bons - A Collction of Zn nd pr-zn Writings compild by Pul Rps, Doubldy, Nw York, 96, p. 4.
F V( ( i d k V 3 ( / k ( k ik On th othr hnd, ħk is th momntum pr qunt chngd with th fild. Wht is lft, formlly t lst, should b th rt of chng of such qunt in d 3 k, nd intgrting this rt ovr k-spc thrfor suggsts tht th totl rt of chng t ll wv vctors is i(/ħv(. Formlly pplying Poisson sttistics nd ponntiting th ngtiv of th pctd numbr of chngs in tim dt thn givs "probbility" P ( i / V( dt tht no chng btwn prticl t nd th fild occurs in tim dt. (Th subscript just indicts tht w r not don dfining th totl probbility yt. Now, w still nd trm tht rprsnts th ffcts of inrti. If th potntil nrgy is msur of th vrg rt t which qunt r chngd with th fild, th mchnicl nrgy could b rltd to intrctions with whtvr it is tht givs prticls thir mss nd inrti. Adding th totl mchnicl nrgy T to th potntil V givs: Pb i(/ ( V( + T( p dt for th probbility of no chng in tim dt. Hr p is th momntum of th prticl nd T(p is th totl (kintic + rst mchnicl nrgy. Now considr prticl strting from loction t tim t with momntum p. Th probbility dnsity for finding it still on its initil trjctory t som ltr tim t will b: i(/ dt'( V( ' + T( p 3 δ ( ( ρc(, t ;, t, v v t t bcus multiplying th probbilitis for ch incrmnt of tim rsults in dding th diffrntil trms in th ponnt. Thy dd up to th pth intgrl of th sum of th potntil nd mchnicl nrgis. Of cours th initil momntum nd position cn t b known simultnously bcus instrumnts tht might b usd to dtrmin thm r lso unprdictbl t th quntum lvl. Nvrthlss, th clssicl mtphysicl principl tht prticl hs uniqu position nd momntum is not ncssrily bndond. Owing to our unvoidbl ignornc, w will just hv to vrg ovr position or momntum or som combintion throf in prcticl sitution. Continuing with th forml ppliction of Poisson sttistics suggsts tht th diffrntil probbility of chng of quntum with wv vctor k in d 3 k during tim dt is d P i( / dt d kv( k k + + p 4 3 d i i(/ ( V( T( dt Aftr picking up momntum ħk, th prticl will b on nw trjctory until th nt quntum is chngd. Th mplitud for rriving t som finl position t tim t cn b prssd s
n intgrl long th initil trjctory of product of thr trms. Th first trm is th probbility for not intrcting long th initil lg, which is givn by P c if w sy th first quntum is chngd t t. Th scond trm is th mplitud for th first chng, which modifis th trjctory th prticl will follow to th scond chng: p p + k, + ( t t v Th third trm is th mplitud for rriving t th finl position t tim t with strting position shiftd to nd t. This rsoning suggsts n intgrl qution ρ(, t;, t, v t i(/ dt'( V( ' + T( p t i(/ dt'( V( ' + T( p t 3 3 ik δ v k ρ t t v t ( ( tt i(/ dt d kv( (, ;,, + v ( tt whr it should b notd tht d 3 kv(k hs th dimnsions of nrgy, nd th ffctiv intgrtion ovr th spc-tim coordints is rstrictd by rltivistic cuslity. In othr words, th intrvls btwn,t nd,t nd btwn,t nd,t must b tim-lik. W cn rwrit th phs fctor tht coms from th Fourir trnsform ccurtly s: ik (/ i ( p p (/ i d p' ' bcus th incrmnt in momntum is concntrtd t th nd of th intrvl whr. This trm combins with th pth intgrl of th nrgy to produc fctor of th form: A i(/ dt'( V( ' + T( p' + dp' ' which pprs to b closly rltd to th Fynmn pth intgrl [3]. In fct, if w intgrt th third trm in th intgrnd by prts onc, w find i(/ dt V( T( i(/ ( p p A ( ' + p' p' v' i(/ dt' L( ', p' i(/ ( p p contins Fynmn s pth intgrl of th Lgrngin L(, p multiplid by i(/ ( p p 3
Now, th vlocity btwn t nd t is v, nd ħk p -p, so w cn rrrng trms to prss th qution in potntilly simplr form: i(/ p ρ(, t;, t, v t i(/ dt'( V( ' + T( p + i(/ p t i(/ dt' L( ', p' t 3 3 i(/ p δ v tt i dt d kv k ρ t t v t ( t t ( t t + v, p p + k, + v ( ( (/ ( (, ;,, This cn b viwd s n intgrl qution for th probbility dnsity for th prticl to rriv t nd t multiplid by phs fctor ssocitd with th (hypothticl initil momntum nd position. W r lft with th pth intgrl of th Lgrngin nd th Fourir trnsform of th potntil to dl with in ctul clcultions. As in Fynmn s formultion, contributions from nrby pths should dd in phs whn th Lgrngin is clos to n trmum nd othrwis cncl. Howvr, only physicl pths r gnrtd by considring quntizd trnsfrs of momntum. Fynmn's formultion includs ll pths in coordint spc long which tim dos not rgrss. This mns tht th prticl is rquird to mov fstr thn th spd of light on som of ths pths. On th othr hnd, th prssion for th Lgrngin tht rsults from our hypothss L V( + T( p p v whil corrct clssiclly, bcus p v is qul to twic th kintic nrgy, lso givs th corrct qutions of motion whn th nrgy nd momntum r corrctd ccording to th spcil thory of rltivity. In fct, th intgrl of th Lgrngin pprs in its Lorntz invrint form ( ( p ( p v T + V dt p d µ It my b tht th inclusion of unphysicl pths in Fynmn's formultion, sing tht it is thn comptibl with Schrodingr's qution, is just vry convnint pproimtion. For mpl in solving th doubl-slit primnt, it is sir to us solutions of Schrodingr's qution thn to comput th pth intgrl plicitly. Howvr, mtching up solutions of Schrodingr's qution in diffrnt rgions forcs th nrgy with which th prticl rchs th scrn to b ctly th sm s it hd pproching th brrir. It sms tht ct consrvtion of nrgy ultimtly follows from th inclusion of unphysicl pths in th pth-intgrl formultion, nd clculting th mplitud s sum rstrictd to physicl pths could giv slightly diffrnt nswr. At lst on mor hypothsis is ncssry in ordr to b bl to prdict th rsults of msurmnts. Sinc our compl probbility dnsity is rltd to Fynmn's pth intgrl, minus th unphysicl pths, th simplst choic is to procd s in stndrd quntum mchnics. Tht, is th msurmnt probbility dnsity should b proportionl to th squr of our compl probbility dnsity. Howvr, th intgrl of this dnsity ovr th sptil coordints might not b constnt in tim if Schrodingr's qution is not stisfid. Thrfor it sms to b ncssry 4 µ
to considr trnsint rthr thn stdy-stt vnts nd intgrt ovr tim to gt th msurmnt dnsity: b. Th probbility dnsity of msurmnt is proportionl to th intgrl ovr tim of th squr of th bsolut mgnitud of th compl probbility dnsity for th systm bing msurd to rriv t th point of th msurmnt. This procdur sms to b rsonbl providd w considr vnts tht r isoltd in tim or sufficintly infrqunt so tht intrfrnc ffcts btwn succssiv prticls in th rcording mdi bcom ngligibl. This, in turn, my impos n primntl uppr limit on th timscl for isoltd vnts in prcticl situtions. Sinc w hv lrdy drivd th Lgrngin, w cn sy tht th clssicl qutions of motion follow from our quntum hypothsis vn if it dprts in som rspcts from th prdictions of convntionl quntum mchnics. Tht thr is som dprtur sms clr from th fct w do not includ unphysicl pths nd do not pct Schrodingr's qution to hold ctly. To clrify whthr or not our quntum principl cn gr with primnt, it sms pproprit nt to invstigt fmilir problm, th doubl-slit primnt. W do find th nrgy is not ctly consrvd. In non-rltivistic css, th chng in nrgy is ngtiv nd on th ordr of th kintic nrgy tht th prticl would hv, ftr scttring, du to its motion prlll to th pln of th slits multiplid by dimnsionlss function of th slit gomtry. (It cn b grtr for nrrow slits. As th rtio of th vlocity prlll to th pln of th slits dividd by th prpndiculr componnt is qul to th scttring ngl, th rtio of th nrgis is proportionl to th squr of th scttring ngl, nd th smll prcntg chng in nrgy might b hrd to dtct.. Quntum Mystry Diffrction by Slits Fynmn usd th doubl-slit primnt s n mpl nd discussd it in grt dtil bcus, h sid, "it contins th only mystry." Th only mystry, of cours, is tht prticls ppr to bhv lik wvs in this primnt. This wv-lik bhvior pprs s intrfrnc frings in th distribution of prticls tht pss through th slits vs. th ngl btwn th incoming nd outgoing momntum of th prticls. Fynmn usd th doubl-slit primnt s pdgogicl tool. Sinc th tim of his lcturs, th pctd intrfrnc bhvior hs bn confirmd dirctly in lgnt primnts [4]. Ect solutions to th qution dvlopd bov r not known yt. Whn th initil pth from long v lds to collision with th scrn, th pnsion my convrg slowly, if t ll, but th rsult must b smll if is on th othr sid of th scrn from. Whn thr is no collision, w my b bl to considr only th lowst-ordr trm. Th compl probbility dnsity tht prticl on trjctory pssing through, t will rriv t,t in this pproimtion is 5
ρ (, t;, t, v tm (, s i( / dt dk dk dk V ( k δ ( v ( t t s tmin (, s i(/ T( tt i(/ ( p p i(/ T( tt 3 y z ( t t ( t t + v, p p + k, + v providd is not on th originl pth. For ch slit s, th trgt position is visibl from th position of th prticl btwn th limits in th intgrtion ovr t. Th intgrtion ovr t includs tims whn th prticl cn scttr bfor pssing through on of th slits on its wy to th obsrvtion point. Tht is, th lowr limit t min is gnrlly lss thn t. Th uppr limit occurs whn th trgt is no longr visibl from th position of th prticl. Th window dos not clos, howvr, if th initil trjctory psss through on of th slits. In tht cs th uppr limit is dtrmind by rltivistic cuslity instd. For simplicity, lt s ssum th prticl initilly pprochs th pln of th slits long th norml nd cll this dirction th z-is. Th potntil is ctully indpndnt of position long th is prlll to th slits, which w ll cll th y-is. Th dpndnc of th Fourir trnsform of th potntil on th corrsponding componnt of th wv vctor will thrfor b concntrtd into dlt function t k y. Thus th outgoing momntum will rmin in th originl pln prpndiculr to th dirction of th slits. Undr ths ssumptions, th first-ordr trm bov simplifis to ρ (, t;, t, v tm (, s i( / h dt dk dk V( k, k δ ( v ( tt p p + k k y s tmin (, s i(/ T( tt i(/ ( pp i(/ T( tt 3 z z Bcus of th dlt functions, it is convnint to convrt th doubl intgrtion ovr dk dk y into quivlnt intgrtions ovr dv dv z. Th Jcobin for this chng of vribls is 6
( k, k k k v v z z ( v, vz k kz vz vz m v z z 3/ 3/ ( v ( v vvz v 3/ 3/ ( v ( v m ( v Th dlt functions tk cr of th intgrtions lving ρ (, t;, t, v (,, v v tm t s 3 i(/ T( tt V( k, kz i(/ ( pp i(/ T( tt δ s min (,, ( tt ( v t t s im ( / π dt ( y y p p + k + v ( tt whr th uppr limit of th intgrtion ovr t cn b st by rltivistic cuslity. This will b th cs if th initil trjctory psss through on of th opnings. Lt s lso ssum th pln from which th slits hs bn cut is vry thin nd th z-dpndnc of th potntil contins dlt function t z. Thn th Fourir trnsform of th potntil is ctully indpndnt of k z so th probbility dnsity cn b writtn ρ (, t;, t, v (,, tm t s 3 i(/ T( tt V( k i(/ ( pp i(/ T( tt δ s min (,, ( tt ( v t t s im ( / π dt ( y y p p + k + v ( tt. Eqution for Sttionry Phs Nt, w nticipt tht contributions to th intgrl will b smll cpt whn th phs of th intgrnd is sttionry. Th sist wy to s whn tht hppns is to not tht + v ( t t 7
This llows th phs in th intgrnd (in units of ħ to b prssd in th prticulrly simpl form φ T( t t + ( p p ( + v( t t + T( tt ( L + p v( t t + ( p p ( + v( t t + ( L+ p v( tt L ( tt p + p + ( L L ( tt This phs is sttionry with rspct to th tim of th intrction whn Th drivtiv of th Lgrngin is givn by dφ dp dl + ( L L + ( t t dt dt dt dl T dv v dt c dt nd th drivtiv of th momntum is rltd to th vlocity nd its rt of chng by dp T dv d + mv ( v dt c dt dt / T dv T dv + v v c dt c v c dt 4 ( / Th condition for sttionry phs cn thrfor b writtn T dv T dv T dv + 4 v v ( L L v ( t t c dt c ( v dt + c dt T dv v v v ( t t + ( L + L c dt ( v c c Th vlocity v ftr th intrction cn b convnintly writtn s 8
v v z v ( t t t t ( + v ( t t ( + v ( t t z z v + v ( t t ( t ( t t t ( t t + v + v ( t t whr w hv introducd th vctor pointing to th finl position from th position +v (tt, which is whr th prticl would b t tim t if it continud on its originl cours without intrcting. (S Figur. for th rltions btwn ths vctors. In turn, th rt of chng of th componnts of v r strightforwrdly clcultd to b dv ( tt v dt t t ( t t ( Thus th rt of chng of v with rspct to t is constnt in dirction, but it divrgs in mgnitud s t pprochs t. W cn us this to rwrit th condition for th phs to b sttionry s v v ( v L L (( ctt ( v c c T ( t t + + If th intrction occurs vry lt, v cn bcom pprcibl comprd to th spd of light c. Othrwis, c(t-t will b lrg comprd to othr quntitis in th qution with th dimnsions of lngth. This mns tht th first trm of th qution will b smll, nd th solution will b found nr L L. Thr will b solution (for fid t bcus th scond trm strts out ngtiv (if w includ tims t bfor t, psss through zro whn nrgy is consrvd ctly, nd thn bcoms positiv for ltr vlus of t. Th condition for sttionry phs cn b rwrittn s / / v v v t t (( ( / v v v ( + c c c c t t v c c c Squring this, nd simplifying somwht, lds to 9
v v +ε +ε + ( c c c tt c ( tt ( v c ( tt c ( tt v v +ε ( +ε + + ct ( t c c ct ( t ct ( t ( v ct ( t ct ( t v ct ( t c c ct ( t ( v ( ( / ( ( ( v +ε +ε +ε + + c c tt c tt c tt b+ε +ε +ε ct ( t ct ( t ( v ct ( t ct ( t ct ( t whr w introduc th dfinitions v v ε ( v c c v v v + ( v c c c b ( v+ v ( t t + + v( t t + + v( t t + for th vctors b, which is nothr vctor tht is constnt in dirction, nd ε, which is smll comprd to b in th non-rltivistic limit. Providd t-t is not infinit, th sttionry-phs condition bcoms quivlnt to ( +ε b+ε v c( t- t ( Th right-hnd sid is lwys positiv, but it will rmin vry smll s long s c(t-t is lrg. Consquntly must b nrly orthogonl to b+ε. Evn if bcoms vry smll, will still hv to b nrly orthogonl to b+ε bcus th right-hnd sid scls lik. If, to form th non-rltivistic limit, w nglct both ε nd th right-hnd sid ltogthr, w hv b ( v( tt + v( t t + + + ( v ( t t ( v ( t t ( t t + v
At this lvl of pproimtion, th phs is sttionry t prticulr vlu of t nd is indpndnt of t. This suggsts tht th pth intgrl of th Lgrngin my not lwys vry s rpidly from its sttionry vlu s is usully ssumd.. Solution Assuming Enrgy is Consrvd Ectly To gt bttr pproimtion, it will b ncssry to hndl crtin smll quntitis dlictly. Anticipting tht nrgy will b pproimtly consrvd, w first not tht v v + c c( t t c t t v + c ( + v ( ( ct ( t ( b v ( ct ( t c + This shows tht squr of th vlocity, nd hnc th nrgy, will b chngd vry littl by th intrction, providd c(t-t is lrg comprd to ny of th quntitis in th problm with th dimnsion of lngth. W cn lso s tht nrgy would b consrvd ctly if wr orthogonl to th vctor b-, which givs vlu of t -t This is quivlnt to ( b t t v t t tt ( tt ( b tt v v which provids on rltion btwn th rmining two fr prmtrs t nd t. Now, if b is qul to, thn it follows tht (b+ε is qul to ( +ε. Using this in th qution for sttionry phs givs ( +ε ct ( t ( v /
if it is prmissibl to divid by ( +ε. Howvr, this rsult is vidntly incomptibl with non-rltivistic vlus of v, so it follows tht both (b+ε nd ( +ε must b zro. Thn, Figur. Dfinitions of Vribls in Doubl-Slit Eprimnt ftr multiplying through by (-v /c [c(t-t ], th qution for sttionry phs simplifis, in th non-rltivistic limit whn nrgy is consrvd ctly, to ( b+ε v v ( ct t ( ct t ( v c( t- t ( ( - ( - b c c v v + + ct ( - t + ct ( - t c c v v v ( ct (- t b + c c c v v v + ( ct (-t b+ + c c c b+ Substituting th vlu for t-t tht corrsponds to ct consrvtion of nrgy, w r ld to th following qution
v v v b v c c c v v v b+ v c c c 4 + + + + On dividing through by th known rsult for[c(t-t ], this bcoms v v v b+ c c c v v v v + b+ + c c c 4 v + v v v v 4 b + + b c c c v v v v b + { + } c c c { } rlting th componnts of whn nrgy is consrvd ctly. Sinc w ssumd is fr prmtr, this condition fis v, or quivlntly v (t-t, s function of. In othr words, it dfins curv of ndpoints of trjctoris for which th phs is sttionry nd nrgy is consrvd. Not tht whn is nrly prpndiculr to v, th rsulting vlu of b bcoms ( vˆ v ( tt v b c c v / v + / v c ( vˆ v ( t t ( v c whr th scond rsult follows if it cn b ssumd tht (. v /c is ngligibl comprd to (- v /c. Rptd us of this pproimtion thn producs n stimt of v (t-t ( t t ( ˆ ( ˆ v v b v v v 3
Th ct rsult is quivlnt to 4 th ordr qution for t-t or quivlntly v /v in trms of (or quivlntly whr v /c nd th componnts of r givn prmtrs. In gnrl, w should pct v to b ngtiv nd trmly smll comprd to b v whn nrgy is consrvd. Furthrmor, b will lso b ngtiv with mgnitud on th ordr of v /c tims -. Substituting into th solution for t -t, w find ( b v( t t vˆ v c / / v( t t v c vˆ / / v ( t t bvˆ v c b v c v c v c + / / which shows tht s bcoms ngtiv nough, th intrction cn occur bfor th prticl rchs th pln of th slits nd still consrv nrgy. In tht cs, of cours, th prticl cn only rch th trgt if it intrscts th pln through on of th slits..3 Uniqunss of Ect Enrgy Consrvtion Aftr multiplying through by -v /c, th qution for sttionry phs cn b rwrittn ( ( ct ( t v b ( b+ε ( b ( b+ε c c(- t t Hr w notic tht cross products of (b+ε nd (b- occur on both sids of th qution, nd by collcting ths trms on th right-hnd sid, w r lft with v ( b+ε + ( b ( b+ε c c(- t t W hv now trnsformd th qution for sttionry phs into n quivlnt on whr th vribl fctor -v /c dos not occur (cpt in th dfinition of ε. Nt, not tht sinc b + v( tt b th trnsformd qution cn b rwrittn such tht ε is ssocitd with v ( +ε v ( +ε + c c( t- t c which givs us qudrtic qution for (+ ε/c(t-t 4
v v v v + ct ( - t + + ( +ε ( + ε ct (- t ct (- t c c c c Th ppropritly smll solution is ( +ε v v v v v v + ct ( - t + ct ( - t ct (- t c c c c c c Howvr, this is quivlnt to / ( b+ε v v v v v v + + ct ( - t + ct ( - t (- (- ct t ct t c c c c c c nd, noting tht (b+ε is rquird to b positiv, w s tht th qution for sttionry phs ctully hs no solution whn th inqulity v v v v v v > + ct ( - t + ct ( - t (- ct t c c c c c c holds. Isolting th rdicl on th right-hnd sid of th inqulity nd thn squring givs v v v v + ct ( - t > (- ct t (- ct t c c c c By multiplying through by[c(t-t ], w s this is quivlnt to v 4 [ ct t ] [ ct t ] v (- + (- + > c c 4 This inqulity obviously holds for lrg c(t-t but would b fls btwn th zros of th qudrtic form on th lft hnd sid. Howvr, th two zros occur t th sm vlu ct (- t v corrsponding to ct nrgy consrvtion, which is thrfor th only solution. / / 5
.4 Sttionry Phs Condition with Trnsvrs Momntum s th Indpndnt Vribl Ect solutions to th qution for sttionry phs r not known yt whn nrgy is not consrvd. W hv trid n pproimtion mthod bsd on smll dprturs from nrgy consrvtion, but unfortuntly th phs dos not ncssrily vry rpidly with smll dprturs from th vlu of t t which th phs is thn found to b sttionry. An ltrnt pproch will b followd in which th indpndnt vribl is chngd from t to k. Subtrcting trms in th phs tht r indpndnt of t-t, w r lft to considr Th first trm cn b rwrittn φ L ( t t + p p + ( L L ( tt T v + c c t t c T v k c c p ( + To crry out th dsird chng of vribls, w cn solv k p T v c mc / { v } c( tt mc v ( v c( t t c( t t c for c(t-t in trms of k. Squring th qution bov lds to qudrtic qution in c(t-t for which th possibl solutions r / ( / ( ( v v mc v c c t t ± / + + ( v v k It s clr tht th uppr (+ sign is th on w nd if th intrction occurs bfor th obsrvtion. Our fundmntl physicl quntity is phs shift prssibl s product of n nrgy nd tim intrvl. Howvr, w my hv th option, first rcognizd by Dirc, of choosing th minus sign nd intrprting th solution moving bckwrds in tim s n 6 / /
ntiprticl. For th problm t hnd, though, w cn us T c k ( t c t k / ( v / v c c mc + / / + + ( / ( / v k v c v c ( v k v mc k + / + + ( / ( / v mc v c v c / / to vlut th scond trm in p. Adding it ll up, w find T v φ L ( t t + p k + + ( L L ( tt c c T v T k + + mc( v c( tt ( v / / c c mc T / v mc T ( / c c k mc k + + mc v c mc mc k mc + + k mc c Sprting out th rdicl trm in th prssion bov for T /c, φ L ( t t + p v mc k ( v / T c ( v ( v mc v k mc mc + mc / + k ( v ( / mc v c v / mc T k v + ( v c k c ( v Finlly, including th rdicl trm plicitly,nd choosing th + sign to gt positiv totl nrgy, th phs cn b writtn 7
φ L ( t t + p ( ( v / c / ( v ( v mc v k mc mc + mc / + k mc v ( v mc v / mc k + / ( v + + ( v c k v mc v k v v mc + mc + v / mc v c ( ( ( / mc v / mc ( v / c k + / ( v + + ( v c k v mc / This prssion is wll bhvd t low momntum trnsfr (ħk <<mc /, nd to scond ordr in k givs φ L( t t + p ( v ( v ( v ( v mc + mc + + ( v mc v ( v k + / + ( v c v mc ( v ( v b+ ( b v ( v k mc + mc / ( v ( / mc v c v ˆ mc v k / + + / ( v c ( v mc k / v v mc On th othr hnd, t high momntum trnsfr (ħk >>mc /, th phs cn b pproimtd by powr sris in mc/ħk 8
φ L( t t + p ( v k ( v ( v mc + mc / + ( v mc v / mc v / ˆ mc k v mc + / ( v / ( v c k mc + + ( v k ( v ( v ˆ k mc v v k mc + + + / / / ( / ( / v c mc c v c v c mc ( v vˆ + mc mc... / + / + ( v ( v In this limit, th phs hs componnt tht is linr in k with lrg cofficint on th ordr of v /c..5 Intgrtion ovr Trnsvrs Momntum Instd of Intrction Tim W will now crry through with th chng of vribl in th prssion for th compl probbility dnsity in th singl-quntum chng pproimtion ρ (, t;, t, v (, tm s iφ 3 V( k δ s min(, ( tt ( v t s im ( / π ( y y dt from t to k. W cn do this by diffrntiting p, which is function of t-t nd t th sm tim qul to ħk / 9
dp T dv T v dv + v dt c dt c ( v c cdt T dv T v v + + c dt c v c c t t t t ( / c( t t ( T v + + c v c c t t v ( / ( c( t t t t T v c ( v ( v v c t t tt Th combintion of fctors in th intgrl tht w nd thrfor bcoms ( ( / dt cdk tt ( v v c T v v c( t t dk c ( t t / m ( v / / ( v ( v c( t t v dk mc m ( v / / ( ( v mc v k + + v k dk / / ( v k ( v ( v m + + v mc / Thus ftr th chng of vribls, th compl probbility dnsity bcoms ρ (, t;, t, v (, v ( v k m s iφ 3 V( k δ / s k min(, s / ( v k ( v ( v m + + v mc im ( / π ( yy dk This cn b usful bcus th limits of th intgrtion ovr k dpnd on whthr th initil
trjctory psss through th slit in qustion nd r othrwis lmost fid vlus (for ch slit..5. Intgrtion ovr Trnsvrs Momntum on Blockd Trjctoris Rfrring to Figur, w s tht th lowr limit of k (s function of corrsponds to th rlist intrction tim t min whr. m v ( t t + vˆ v ( t t + vˆ min This rsults in th prticl intrscting th pln t th top ( m of slit s. Solving for v (t-t min givs v ( t t v t t + vˆ ( m min m m ( v ( t t vˆ + v m v v m Assuming th initil trjctory dos not pss through th slit, similr rltion ists btwn th mimum vlu of t nd th position of th bottom of th slit ˆ ˆ ˆ Figur. Dfinitions of Minimum nd Mimum Intrction Tims
v ( t t v t t + vˆ ( ( vˆ m min vˆ vˆ min It thrfor turns out tht whn v is smll comprd to v ( w shll s tht it is vry smll, th -componnts of th vlocity t th two limits r smll nd pproimtly indpndnt of. v m v min ( v ( min v vˆ vˆ vˆ ˆ v min min ( v vˆ min ( m v vˆ vˆ m ( v vˆ m Ths vlocitis r, howvr, slightly diffrnt for ch slit. In th sm pproimtion, th componnt of th vlocity long th initil dirction rmins ssntilly qul to v. This mns tht th momntum trnsfr is smll nd nrly constnt t k mv ( + / mv s s mv min m s vˆ ˆ v It will b noticd tht th diffrnc in mn momntum for th two slits is smll in proportion to th distnc from th is of symmtry to slit position comprd to th trgt position k k s s whr k -br corrsponds to scttring on th is of symmtry in th pln of th slits. Th width of th rng of th intgrtion will b
( δ k m vm vmin m min mv vˆ w mv vˆ nd its mgnitud comprd to th mn momntum trnsfr δ k w k s s will lso b smll if th slit gomtry is comprssd in ordr to mk th first minimum occur t lrgr vlu of whr it should b sir to obsrv. Of cours, th limiting nd mn vlocitis r slightly diffrnt for ch slit, so th compl probbility dnsity bcoms pproimtly ρ (, t;, t, v v v k m(, s iφ i( / π δ( yy V( ks dk / s k min(, s / ( v k ( v ( v + + v mc In typicl cs, th mn vlus of momntum trnsfr for th two slits r not much diffrnt, so w gt th pctd form tht is proportionl to th Fourir trnsform t th wv vctor tht corrsponds to scttring in th vicinity of th slits. To scond ordr in ħk /mc nd v /, but cluding trms of scond ordr in both t th sm tim, th phs cn b writtn ( v φl ( t t + p mc + k ξ + σ k whr w hv introducd th dfinitions ( v ( ( v { } / ξ ξ σ k v mc + + / σ σ 3
ξ σ ( v b+ ( b v ( v ( v v / / ( v c mc Not tht, to th tnt tht v is smll comprd to v, th cofficint of th qudrtic trm in k is pproimtly indpndnt of. Th phs is sttionry cntrd t momntum trnsfr / k mc sttionry ξ mc σ ( v c / / ξ v Th linr trm in th phs bcoms significnt whn k ξ is of th ordr of unity or lrgr. In non-rltivistic primnt, th qudrtic trm in th phs is likly to b smll on blockd trjctoris, s cn b sn by rrrnging ( σ k v mv ( v v mv s / mc vˆ s s / s ˆ s c v ( v c / Th scond fctor would b bout ±π/4 if corrsponds to th first minimum t th trgt. Supposing w hv chosn to b bout tims s -br, in ordr to mk th first minimum mor sily obsrvbl, th first fctor will b lss thn (v /c nd would b smll if v /c is lss thn bout -. Th phs will vry rpidly t vlus of momntum trnsfr whr rltivistic ffcts r significnt, tht is if ħk /mc is not ngligibl comprd to unity. Howvr, th intgrnd rmins boundd bcus th Fourir trnsform of th potntil contributs fctor of k in th dnomintor 4
V( k kv( k / ( v / k mc c k v ( / v c + + v mc mc kvk ( v mc ( which long with th fctor tht ws lrdy thr, combins with th Lorntz fctor (-v /c to giv v /c which pprochs unity in this limit. Thus, for th blockd trjctoris, th compl probbility dnsity bcoms pproimtly ρ (, t;, t, v v v k m(, s V( k iφ i( / π δ( yy dk s 3/ s ( v k min(, s ( v ( i/ L ( tt p + mc v / c k ik ξ + / ( v mc / ( v k m (, s V( ks i( / π δ( yy dk 3/ v s k, s ( min( Not tht th contribution from trjctoris pssing though ch slit s r sprtly proportionl to V(k s -br, nd ch shows th minimum tht is ttributd to dstructiv intrfrnc in convntionl quntum mchnics. In typicl css, th trnsform of th potntil is lmost th sm for trjctoris pssing through th two slits, so thir contributions do not ncssrily cncl or intrfr. Noting tht whn th qudrtic trm is ngligibl ovr th rstrictd rng of momntum trnsfr on blockd trjctoris, w cn vlut th intgrl ovr dk v / c k km(, s ik ξ + / k m(, s ( / mc v c ik { ξ+ σ k } dk dk kmin(, s kmin(, s k m(, s ik ξ dk k min(, s ik (, s ξ ik (, s ξ δ k m min ik sξ iξ sin δkξ / δk ξ / 5
.5. Intgrtion ovr Trnsvrs Momntum on Unblockd Trjctoris If, on th othr hnd, th initil trjctory dos pss through slit, th uppr limit of th intgrtion ovr k bcoms infinit. Rlvnt to tht limit, th Fourir trnsform of th potntil + V( k dv( π V δ ( π ik ik V k d slits L/ L/+ W ik V Vδ ( k + π ik L/W L/ V Vδ k + k L + W k L { } ( sin ( / sin ( / kπ contributs nothr fctor of k in th dnomintor. W hv ssumd tht th dlt function is outsid th rng of th intgrtion ovr k, so w hv ρ(, t;, t, v V { sin k ( L/ + W sin k ( L/ } im yy dk iφ 3 ( / π δ( / s k min(, s / ( v k π m( v ( v k + + v mc At high momntum trnsfr, th dnomintor of th intgrnd scls lik k (-v /c (mv /ħ, which pprochs constnt vlu bcus v is boundd by c. Also in this limit, th phs is gin pproimtly linr in k, but its cofficint is on th ordr of.v /c, which is lrg in mgnitud comprd to L/+W nd L/ in th css of primry intrst. Thrfor, th intgrnd will oscillt rpidly nd contribut littl to th compl probbility dnsity. In typicl css, ħk min /mc is on th ordr of ( k mc v min m vˆ c ssuming tht v is smll comprd to v. On th othr hnd, bcus w r intrstd in th first null in th diffrction pttrn, 6
L+ W W L+ W W sin k + sin k sin k( L/ + W sin k( L/ k k L+ W W cosk sink k it follows tht k min (L+W/ will b of th ordr of unity, or mor spcificlly π/. Thrfor, whn th rltivistic corrctions bcom significnt, k will b mny ordrs of mgnitud lrgr thn k min, nd th intgrnd will b mny ordrs of mgnitud smllr. Thrfor, oscilltions t smll but finit mplitud t high momntum trnsfr cn b ignord in first pproimtion nd w cn considr ρ (, t;, t, v V i( / π ( y y dk δ s π v c 3/ ( / k min(, s ( v ( i/ L( tt p + mc / ( v V dk δ 3/ v π s k k min, s i(/ π ( y y sin k ( ( { sin ( / + sin ( / } k L W k L { ( ( } { ik } ξ+ ο k L/+ W sin k L/ It is usful to tnd th lowr limit of th intgrtion to nd thn considr th compnsting intgrl ovr th finit limits sprtly ρ (, t;, t, v ( v ( i/ L( t t p mc dk + / iο k ( { sin k ( / sin ( / } / L W k L v c + V k + isin kξ δ 3/ k min(, s s ( v π dk L+ W W ik { ξ+ ο k} cosk sink k i( / π ( y y k iφ cos k ξ ik ξ+ ο k sin ( / sin ( / ( / k L + W k L v c k ( i/ L ( tt p + mc / ( v V dk 3/ ( ( ( v π s k k min(, s dk L W W ik { ξ+ ο k + } cosk sink k { } { } { }( i(/ π δ( y y + i sin k L/+ W sin k L/ sink ξ which is possibl bcus th intgrnd is wll bhvd t k. Th rl prt of th infinit intgrl rproducs th potntil vlutd t ξ with som loss of high sptil frquncis du to th qudrtic trm in th phs. If this loss cn b nglctd, tht is σ cn b st to zro, w thn hv 7 dk iο k
L L { ( ( } { } W ξ dk ik sin ξ+ ο k π < < k L/ W sin k L/ + k L L < ξ < + W In othr words, this prt of th rsult is non-zro constnt whn ξ, which hs th dimnsion of lngth, is in on of th slits. If w st σ qul to first, th rmining imginry prt of th infinit intgrl is i k L W k L k ( L/ + W + ξ( L/ξ ( + ξ( + ξ dk i { sin ( / + sin ( / }( sin ξ ln k 4 L/ W L/ ccording to Grdshtyn nd Ryzhik [5] 3.74 # (pg 44. This rsult hs logrithmic singulritis whr ξ coincids with th dgs of th slits, but it bcoms rpidly smllr s ξ movs wy from th slits. For th tim bing, it will b ssumd tht this form givs corrct rsults ftr th intgrtion ovr d. Finlly, if th qudrtic trm in th phs cn b nglctd in th infinit intgrls ovr dk, it will b rsonbl to nglct it in th intgrl from to k mins s wll ρ (, t;, t, v L L W ξ π < < L L < ξ < + W ( v ( i/ L( tt p + mc / ( v V i ( L/ + W + ξ( L/ξ i( / π δ( yy ln 3/ ( 4 ( / ( / v π L W ξ L ξ s + + k min(, s dk ik cosk L W sin k W + ξ k.6 Intgrting Ovr th Bm So fr, w hv clcultd th compl probbility dnsity rsulting from singl prticl ntring with vlocity v long th z-is on trjctory pssing through som fid. Of cours, w don t know how to prpr tht initil stt. Most likly, w would hv distnt point sourc, nd th displcmnt of th initil trjctory from th z-is would b unknown. Bcus of th trnsltion invrinc of th boundry conditions long th y-is, intgrting th probbility dnsity ovr y will rplc th dlt function δ(y-y by N y which is th dnsity of prticls in th y-dirction. Likwis, dding up th contributions in th -dirction ntils multiplying by fctor of N s wll s intgrting ovr d. Th finl rsult, of cours is thn indpndnt of. 8
.6. Intgrting Ovr th Bm on Blockd Trjctoris For th bov rsons, for blockd trjctoris, th vrgd contribution bcoms ρ (,; tt, v ( v ( i/ L( tt p + mc / ( v mins d ik k sξ δ y 3/ s δ s v δk i( / π N N V( k k ( sin ξ / ξ / Th intgrl ovr d cn b put in mor stndrd form by chnging th vribl of intgrtion to ξ,, which givs ( bv ˆ ( v ˆ d dξ + ( v ± ( ξ + / ( bv ˆ ( v ˆ ( v c / d It lso follows from th dfinition of ξ tht th phs trm k ξ of th intgrnd incrss monotoniclly with ovr th rng (-, in which is positiv if ( b vˆ ( vˆ ( vˆ v ( tt v c v c > / / ( Othrwis, (tht is if. v < ξ psss through mimum vlu ξ m t ( sp ξ m sp t th vlu of sp (for sttionry phs givn by ( bv ˆ ( v ˆ / sp v c / ( vˆ v ( tt v Figur 3 shows mpls of k ξ plottd vs / t diffrnt vlus of sp / ssuming th incoming prticl is non-rltivistic nd s -br/., v /. v -5 nd mv /ħ.( 7. Solving for. v, w find / 9
( vˆ vˆ( vˆ ( v c ( sp + / > (( ( v c ( sp ( v ( sp / vˆ vˆ > vˆ + vˆ + / Roughly, th minimum vlu of - v /v is s smll comprd to - sp s - sp is to v /v, It will b notd tht ct consrvtion of nrgy on th initil trjctory dfind by sp cn b in this rng. Undr th ssumption tht (. v /c is ngligibl comprd to (-v /c, nrgy is consrvd if v sp < v Of cours, th initil trjctoris tht contribut th most to th compl probbility dnsity lso corrspond to smll vlus of ξ m sp. Upon th chng of vribls, th trm tht thn pprs in th dnomintor, instd of, upon intgrting ovr dξ instd of d cn b vlutd by solving th dfinition of ξ s qudrtic qution for, ( bv ˆ ( v ˆ ( ξ ± ( ξ + / ( v c / giving ( ( ( ( bv ˆ v ˆ bv ˆ v ˆ + ± ( ξ + ( v v ( / For positiv vlus of th positiv sign is rquird whn v is positiv but whn v is ngtiv, th ngtiv sign is ndd in th rng sp < < mins so d sp dξ sp ( bv ˆ( v ˆ ( ξ + ( v / nd th nt rsult of chnging th vribl of intgrtion is 3
Figur 3 Non-rltivistic doubl slit primnt with s -br/., v /. v -5 nd mv /ħ.( 7. ρ (,; tt, v ( v ( i/ L( tt p + mc / ( v ( / π y ( 3/ s δ s i N N V k k ξ ( ( v mins sp dξ ik sin / sξ δk ξ / ( ( / sp bvˆ ˆ v δk ξ ( ξ + ( v If sp is in th rng of th intgrtion th intgrl ovr dξ, it will b th mimum vlu rchd by ξ, nd th intgrl cn b considrd in two prts 3
ξ ( mins sp dξ ik sin / sξ δk ξ / ( ( / sp bv ˆ ˆ v δk ξ ( ξ + ( v sp sp dξ ik sin / sξ δk ξ dξ / + ( ( / bvˆ ˆ v δk ξ ξ ( mins bv ˆ ˆ v ( ξ + ( ξ + v v ( ( ( ( / ik s ξ sin δkξ / δkξ / If, in ddition, k s -br is much lrgr thn δk (mking th first minimum t th trgt mor sily obsrvbl th phs trm k s -br ξ will bgin to oscillt rpidly bfor δk ξ/ bcoms significntly diffrnt from δk s sp /. In such css, th intgrl ovr dξ bcoms pproimtly ξ ( mins sp dξ ik sin / sξ δk ξ / ( ( / sp bv ˆ ˆ v δk ξ ( ξ + ( v sp ik sin / sξ sp ik sξ δ k sp dξ dξ / δ k / + sp ( bv ˆ( v ˆ ξ ( mins bv ˆ ˆ v ( ξ + ( ξ + v v ( ( ( ( Ths intgrls cn b put in mor stndrd forms by chnging th intgrtion vribl to ( ξ ( bv ˆ ( v ˆ ( v c ( ξ ( sp + / nd choosing th positiv rng for th vribl ν rsults in ( ξ ( sp ( sp ( sp { v } / 3
ξ ( mins sp dξ ik sin / sξ δk ξ / ( ( / sp bv ˆ ˆ v δk ξ ( ξ + ( v sp ik sξ sp ik sξ sin δkξ / dξ dξ / δkξ / + ( bv ˆ( v ˆ ξ ( mins bv ˆ ˆ v ( ξ + ( ξ + v v ( ( ( ( ik ( ( ( min / s sp ν ξ s ( sp iks ( sp ν sin δ k / sp ik s dν dν / / δ ksp / + ( ν ( ν iks ( sp sin k ν δ sp / iks ( sp ν ik s dν dν / / δ ksp / ( ν ( ξ ( min /( ( s ν sp Th first intgrl is rltd to Hnkl function, s Grdshtyn nd Rhzhic [5] (pg 956 #6, so w cn writ ξ ( mins sp dξ ik sin / sξ δk ξ / ( ( / sp bvˆ vˆ δk ξ ( ξ + ( v sin δ ksp / π ik ( ( ( s ξ mins H ( ks ( sp H, ks ( sp δ ksp / i sp whr th scond trm in th curly brckts rprsnts n incomplt Hnkl function dfind by stting th lowr limit of th intgrtion to som positiv rl vlu tht is grtr thn. Th contribution to th compl probbility dnsity from th blockd trjctoris in this cs is ρ ( v ( i/ L( tt p + mc / ( v ( tt v i π NN y A 3/ BS ( sp Vks s v,;, (/ ( ( ABS sp H ks sp H, ks ( sp sp / i sp sin δ ksp / π ik ( ( ( s ξ mins ( ( ( sp mins whr th cofficint A Bs hs bn introducd for comprison with th ffcts of th unblockd trjctoris. Whn, on th othr hnd, sp is grtr thn mins w will hv / 33
ξ ( mins ξ ( sp dξ ik sin / sξ δk ξ / ( ( / sp bv ˆ ˆ v δk ξ ( ξ + ( v dξ sin δk ξ / mins ik sξ / ( ( / bvˆ ˆ v δk ξ ( ξ + / ( v c th uppr limit of th intgrtion is rchd bfor th phs trm k -br ξ bcoms sttionry with rspct to. Howvr, it psss through zro bfor th uppr limit is rchd, so th sin of δk ξ/ dividd by δk ξ/ is sttionry t points strddling th origin nd vris rthr slowly btwn thm. Thn bcus th phs trm k -br ξ oscillts incrsingly rpidly with s it rcds from th origin, ξ ( mins sp dξ ik sin / sξ δk ξ / ( ( / sp bvˆ vˆ δk ξ ( ξ + ( v ξ ( mins sin δkξ( mins / dξ ik sξ / δkξ( mins / ( bv ˆ( v ˆ ( ξ + ( v sin δkξ( mins / dν iks ( ( sp δkξ( m / in / s ( ξ ( min /( ( s ν sp sin δkξ( min s / π ik ( ( s ξ min s H, ks ( sp δkξ( mins / i sp Of cours, ξ( mins is lss thn sp nd rchs mimum t sp s long s th lttr is lss thn. Th cofficint in this cs is thrfor ( ( sin δkξ mins / π ik s ξ mins ABS H, ks ( sp ξ mins / i sp ( (.6. Intgrting Ovr th Bm on Unblockd Trjctoris For unblockd trjctoris, ssuming th qudrtic trm in th phs cn b nglctd, w hv 34
ρ (, tt ;, v L L W ξ π < < L L < ξ < + W ms d i i( / π ln π 4 L/ W ξ L/ ξ ( v ( i/ L( tt p + mc / ( v V NNy 3/ s v ( ( L/ + W + ξ( L/ξ ( + ( + mins k min(, s dk cos L + ik k W sink W ξ k Th first two trms will not contribut unlss ξ, which is function of, tks vlu nr on of th two rngs dfining th slits for vlus of lso in on of th rngs. Sinc is (probbly chosn to b smll comprd to in such rngs, w cn us linr pproimtion for ξ ξ b + ( bv ˆ ( v ˆ ( / v c ( bv ˆ ( v ˆ ( v ( / ξ ( / + + + Assuming is bout n ordr of mgnitud lrgr thn in th rlvnt rngs, w s tht ξ will not vry much during th intgrtion ovr d, but in ordr to b in on of th llowd rngs t ll, it must b tht v stisfis vˆ ξ ( v bv ˆ for som ξ such tht ξ (+ / nd ch fll within th rngs dfining on of th two slits. Thus for th trms undr considrtion to ctully contribut, w s tht v /v must b ngtiv nd bout s smll comprd to s is comprd to b v /v. If tht is th cs, w will hv ms mins L L L L W ξ W ξ d π < < < < πw L L ( s / L L < ξ < + W < ξ < + W providd ξ (+ / rmins within on of th slits whil vris ovr th rng ( mins, ms. Othrwis, th rsult will b smllr going to zro if ξ (+ / rmins outsid both slits ovr 35
th full rng. Th scond trm contributs two intgrls of th form ( + ( b+ c ln + ( b+ c ( + ( b+ c ( ( + ln ( + + ( ( + d ln ( ( b + c b b c b c b c which shows tht th logrithmic singulritis of th intgrnd will b cncld. Using bξ /, w cn writ L+ W W ms i d ξ + ln s (,,, 4 ii L W ξ mins L+ W W ξ W L+ W W L+ W W L+ W + ξ + / ln ξ / ξ / + + + + L+ W W L+ W W L+ W W L+ W i ξ + / ln / / ξ + + ξ + ξ ( s / W L+ W W L+ W W L+ W + ξ / ln + ξ / + + ξ / W L+ W W L+ W W L+ W + ξ / l n ξ / ξ / ms mins This still pks nr vlus of ξ corrsponding to th dgs of th slits s cn b sn in Figur 4, which ssums th slits hv qul lin nd spc gomtry nd (L+W/ quls.. Of cours, ths pks would b smoothd down if th qudrtic trm in th phs hd not bn st to zro. Morovr, th contributions from th uppr nd lowr slits pproimtly cncl cpt nr thir dgs. For smll vlus of ξ, it is usful to rwrit th third trm in th form ms mins k min(, s ms k min(, s ik ξ cos sin cos d dk L+ W W W L+ W k k d dk k k ik which is pproprit in th limit tht k mins W/ is smll comprd to unity. Actully, sinc w r intrstd in th first minimum t th trgt, k mins (L+W/ will b bout π/. Thn, vn in th cs tht th width of th slits is qul to th spc btwn thm, k mins W/ would b only π/4, whr th sin is still pproimtly linr in its rgumnt. Nglcting th smll dpndnc of k mins on, on cn intrchng th ordr of th intgrtions to gt mins ξ 36
(, ms k min s d L+ W ikξ( + / dkw cos k mins kmins ms L+ W ikξ d ikξ( / dkcos k W mins k mins ik ikξ( / ms ξ L+ W W dkcos k ( s / ikξ / min s kmins W L+ W ikξ + s/ ξ dkcos k ( s ξ ( sin k ( W / ( / k W / Evn if ξ is s lrg s, k ξ W/ will b smll in th currnt ssumptions bout k min nd W, so th intgrtion ovr k cn b pproimtd by ms mins (, k min s + cos kmins W L+ W cos s d L W dk W k dk k / ( ξ ( + / ik ξ ( + / ik s L+ W L+ W ik + ξ( + s/ ik + ξ( + s/ W + ( s / L+ W L+ W i + ξ( + s / i + ξ + s / L+ W L+ W ik min s + ξ( + s/ ik min s + ξ( + s/ W + i( s / L W L W + + + ξ( + s / + ξ( + s / ( kmins This form is wll bhvd t smll vlus of ξ. Also, it is lmost th sm for both slits, sinc s -br is lwys dividd by. Evn whn ξ is t th cntr of on of th slits ±(L+W/, nd k (L+W/π/, w will hv ms mins k min(, s d L+ W W W W dk W k k i ± / L+ W ikξ ( + / cos min ( s At ths prticulr vlus of ξ, w find th rl nd imginry prts of th third trm prtilly cncl th first nd scond trms, rspctivly. Whn ξ ±(L+W/ r both lrg comprd to W/, it is usful to rwrit th third trm in th form 37
ms mins ms mins ms mins k min(, s d dk L+ W W cosk sink k k min(, s d dk L+ W W L+ W W sin k sin k + k d ik k min(, s L+ W W L+ W W L+ W W L+ W W ik dk ξ + ik ξ ik ξ ik ξ + + + ik ξ L+ W W L+ W W L+ W W kmin(, s kmin(, s kmin(, s ξ+ + ξ ξ+ min( ms d du iu du iu du, L+ W W k s ξ + iu mins i u i u + i u ξ ik + Thn if ξ flls outsid both slits L + W W ξ ± > th intgrls combin in pirs to giv ms mins Figur 4 Imginry prt of th infinit intgrl ovr momntum trnsfr d dk d k i u i u du i u L+ W W L+ W W k min ( (, min(, min, s k s k s ξ+ + ξ + ms L+ W W ik ξ du iu du iu cosk sink + mins L+ W W L+ W W kmin(, s ξ+ kmin(, s ξ iu 38
Furthrmor, in th limit L + W W ξ ± th vrition in th dnomintor bcoms ngligibl ovr th rngs of th intgrtions giving ms mins mins k min(, s d dk L+ W W cosk sink k L+ W W k min(, s ξ + + m s d iu du + ξ ik L+ W L+ W i k (, s ξ + ( i k, s ( ξ ( min(, L+ W W k s ξ + L+ W W L+ W W min k min, s ξ + min k min, s ξ Nglcting lso th smll vrition of k min (,s on nd collcting trms givs ms mins k min(, s d dk L+ W W cosk sink k ik ms ikmins L/ W ikmins L/ d ξ + ξ ikmins ikmins d + d L W L W mins + i ξ L/ + i ξ L/W + ms ikminsξ L/ + W L/ d ikmins L+ W ikmins ikmins ξ d d d mins L W + slits L/ L/W i ξ W hv sn tht th first two trms in th compl probbility du to n unblockd trjctory r only significnt whn ξ is in th vicinity of th slits. Thus, s th mgnitud of ξ incrss so tht ξ ±(L+W/ both bcom lrg comprd to W/, th contribution from unblockd trjctoris bcoms ξ du iu 39
ρ ( v ( i/ L( tt p + mc / ( v (, tt ;, v i(/ π NN y 3/ s ( v V + + π ξ i ms ikminsξ d i ik sin mins L + W kmin s ( sin ξ d iw L W kmins mins L+ W slits kminsw π ( / N N y s ( v ( i/ L( tt p + mc / ( v ( v 3/ ms ikminsξ d iξ V ( k min s mins L+ W ξ ms ik min sξ d W V ( L+ W L+ W sin kminsw / sin k mins mins k L+ W π minsw / ξ W / / Ecpt for th ponntil trm, th intgrnds r lmost constnt ovr th smll rng of vlus of in both css. Th vrg vlu of ξ in prticulr slit s is ( bv ˆ ( v ˆ ms ξs d b W + ( min / s v c ms d ( / W ξ + mins ms + mins s ξ+ ξ+ Using this nottion, th pproimt rsult cn finlly b writtn 4
ρ (, tt ;, v i ( / π N N y s ( v ( i/ L( tt p + mc / ( v ( v 3/ ikminsξ s i Wξs sin kminsξw / V ( kmins s k mins W / L W ξ + ξs ikminsξ s W ( L+ W / sin kminsξ W / V L+ W sin kminsw / sin k mins s k mins W / kmins / L W ξ π + ξs For comprison of th two trms it is usful to isolt th combintion of fctors nd dfin th complmnt of th Fourir trnsform V C by C V L+ W sin kminsw / V ( kmin s sin kmin s π kmins / V ( cos kmin s( L/ W cos kmin s( L/ + kminsπ which hs th sm mplitud s th Fourir trnsform of th potntil but, bing constructd of cosin trms instd of sin trms, is substntilly out of phs with it. Thn introducing A U nd B U, s th cofficints of th Fourir trnsform nd its complmnt in th contributions from th unblockd trjctoris, w cn writ ρ (, tt ;, v ( v ( i/ L( tt p + mc / ( v i( / π NNy A 3/ Us ξ V k mins BUs ξ V k mins s v ( { ( ( ( ( } + Th rtio of th two cofficints B A Us Us i ( L + W ξ is smll (in bsolut mgnitud to th tnt tht ξ is lrg in comprison with th distnc (L +W/ from th is of symmtry to th cntr positions of th slits. By th wy, ξ is rltd to sp by 4
( bv ˆ ( v ˆ ξ + v ( ( sp nd w cn s tht if - sp >>3 / thn ξ will b lss thn -. This mns tht if nd - sp r both lrg comprd to {L+W/, ξ will b lso..7 Probbility Dnsity of Rcordd Msurmnts Whn ξ is ngtiv nd lrg in mgnitud comprd to (L+W/, th compl probbility contributions from th blockd nd unblockd trjctoris r both proportionl to th Fourir trnsform of th potntil with cofficints for th blockd trjctoris nd sin δ k / A H k ( ( sp π ik s ( ( Bs sp s sp sp / i A Us ikminsξ s i Wξs sin kminsξ W / s k mins W / L+ W ξ ξs for th unblockd trjctoris. It will b shown tht th contributions of th blockd trjctoris domint whn is chosn to b t th first minimum of th diffrction pttrn, tht is k (L +W/ is qul to π/, nd lso hppns to b lrg comprd to (L+W/ which in turn is lrg comprd to W. Considring th blockd trjctoris first, it is notd tht th rgumnt of th Hnkl function is rthr lrg, so w cn us th pproimtion (Grdshtyn nd Rhzhic [5] s 8.45 #4 on Pg 96 / iks sp iπ /4 ( H ( ks ( sp π ks ( sp Thrfor, th cofficints for th blockd trjctoris bcom ( 4
( sp sin δ k / π ( sp iks sp iπ /4 ABs sp / i π ks sp sin δ k / sp / i s π k ( sp s ik sp iπ /4 sp π / / whr w hv usd k mv k s s s vˆ to dfin k -br. Nglcting th smll slit dpndnc of th mplituds, th ffct of th phs shift btwn thm bcoms / sin δ ksp / π iksp iπ /4 ik( L+ W sp/ ik( L+ W sp/ ABs ( sp ( ( + s sp / i π ks sp / ( sin δ k / π L+ W δ k k ( sp ikspiπ /4 sp cos δ ksp / i π k s sp This nvlop of this prssion dcrss slowly s sp rtrts from - to lrgr ngtiv vlus s long s δk sp / rmins smll w δ ksp / k sp / L+ W W sp π k < L+ W In prticulr, if is chosn to b th first null in th diffrction pttrn, thn this condition is quivlnt to sp L + W < W Thus th rng of (ngtiv vlus of sp in which th sum of th cofficints for th blockd trjctoris cn b significnt is lrg (comprd to to th tnt W/(L+W is smll. Within this rng, th sum of th cofficints for th blockd trjctoris bcoms ( ( L ikspiπ /4 sp ABs sp iδ k cosk s k s sp / π + W ( 43
Turning now to th blockd trjctoris, thr is lso phs shift btwn th contributions from th two slits. Actully, both trms in th phs vry nd k ξ k ξ ms s kξ + s W s kξ + W W s s kξ ms mins s s Avrging th smll diffrnc in th mgnituds of th two trms, th nt contribution from th two slits du to unblockd trjctoris is givn (to lowst ordr in W/ is givn by s W L W + ik ξ i Wξ sin kξw / W L + W AUs cosk ξ k L+ W ξw / ξ Th nvlop of this prssion dcrss lik (ξ - s ξ rtrts from - towrds mor ngtiv vlus s long s k -br ξ W/ rmins smll L+ W W ξ kξw / k L + W From its dfinition, w cn s tht ξ ( bv ˆ ( v ˆ ξ + v ( ( sp sp sp is pproimtly qul to sp whn sp is smll comprd to. As long s ξ is lrg comprd to (L+W/, but lss thn bout (+L/W, th nt contribution from th two slits du to th unblockd trjctoris will b pproimtly 44
s i W L+ W W L+ W ik ξ Wξ AUs cosk ξ L+ W ξ At lrgr vlus of sp on th ordr of, ξ bgins to grow qudrticlly. This mns tht s sp rtrts from - towrds lrgr ngtiv vlus, th contribution from th unblockd trjctoris will cut off bfor tht of th blockd trjctoris dos. For mpl, whn - sp is lrgr thn 3 /, ξ will b lss thn -. Th nvlop of th blockd trjctoris bgins to dcrs lik (ξ - ~ ( osp -4 whn ξ / < -(L+W/W which occurs whn sp L+ W < + W If (L+W/W is lrg, th rng in sp in which th blockd trjctory contributions rmin significnt ftr th unblockd trjctoris hv cut off / L L + < sp < + + W W Evn bfor unblockd trjctoris cut off, for mpl whr ξ - th pk vlus of thir mplitud / W ξ k( sp A δk L+ W π ξ / ξ k ( sp k L+ W π ξ sp L+ W ξ / L+ W L + W ( ξ / π k AUs s Bs ( sp s / / is smll comprd to tht of th blockd trjctoris. Tht is, t th first minimum of th Fourir trnsform of th potntil πk -br(l+w/ is bout π. Also, still ssuming is tims (L +W/, th rtio of th mplituds will b on th ordr of - tims function of th rducd vribl ξ / (or sp /. For mpl, whn - sp is 3 / giving ξ -, th function is bout 3 /4 nd th blockd trjctoris r still dominnt. 45
By hypothsis b, th diffrntil probbility of rcording th prticl t in tim dt whn - sp is lrgr thn 3 / is thrfor pproimtly ρ (,; tt, v dt / y sin δ / sp π ( + sp s NN k L W ( / π V( k δk cos 3/ k dt ( v / ( s δ k sp k s sp Nt considr th chng of vribl which thn lds to ρ (,; tt, v dt ( v c ( π ( ( bv ˆ ( v ˆ / sp v c ( vˆ v ( tt ( / v v t t dt d sp sp v NN ( k δ k ( L+ W / ( cos v / / y π δ sin sp / sp V k 3/ s k dsp v ( t t k c s δ k sp / Th chng of intgrtion vribl from dt to d sp cncls th fctor of - sp, lving rltivly simpl function t th lowr limit of th intgrtion. Th intgrnd hs zro t δk sp / -π nd rmins smll for sp / lss thn bout -π(l+w/w. It lso hs zros whr sp / is ngtiv odd intgr ssuming tht th first null in th diffrction pttrn is t. Strictly spking, th fctor of t-t is not constnt, but solving for it in trms of sp ( ( vˆ + ( / ( sp / ( ( ( v ( sp v t t v c + w s tht th intgrnd will bcom vry smll bfor v (t-t bcoms significntly diffrnt from. It's pprnc in th dnomintor is pproprit for th -dimnsionl primnt bing considrd. Whn sp pprochs zro from ngtiv vlus, k (- sp rmins lrg, so th compl 46 / /
probbility contribution from th blockd trjctoris pprochs vlu tht hs th sm functionl form on sp but is dividd by. Howvr, w find ξ ξ sp convrgs to sp. This mns tht th componnt of th compl probbility dnsity du to th unblockd trjctoris tht is in phs with th Fourir trnsform of th potntil dcrss linrly whil th out of phs componnt rmins substntilly constnt. Onc sp turns positiv, though, th contribution from th blockd trjctoris will dcrs s th lowr limit of th incomplt Hnkl function incrss from unity towrds lrgr vlus whr th intgrnd is smllr in mgnitud. Th unbound trjctoris my thn domint s sp pprochs, which is th uppr limit of th rng ovr which th phs is sttionry t som vlu of k, ξ pprochs /. At this uppr limit, th contribution from th unblockd trjctoris will gin b ssntilly proportionl to th Fourir trnsform ssuming / is lrg comprd to (L+W/. Howvr, th fctor of - sp rsulting from chnging th intgrtion vribl from t to sp will thn cncl this rmining contribution from th unblockd trjctoris. On th othr hnd, th rng of rrivl tims t which th contributions from th blockd trjctoris cn b significnt corrsponds roughly to sp π δ k sp / Sinc δk is roughly W/ smllr thn k, this mns th lowr limit of sp is on th ordr of -π/k W or sp ( L+ W W whn th first minimum is t. Thrfor - sp cn b lrg comprd to if W is smll comprd to (L+W/ nd still s lrg s 5 for W qul to L. Thus it mrgs tht th blockd trjctoris cn b significnt rltivly fr from th slits, comprd to th obsrvtion point, nd thrfor contribut much mor thn th unblockd trjctoris. W cn now stimt th chng in th nrgy of prticl on trjctory dfind by rriving t th trgt position t tim t givn in trms of sp / k ( v / v mc c k T T + / + + ( / ( / v mc v c v c k v k + / ( v ( v m 47