Toward a Foundational Principl for Quantum Mchanic Th wind i not moving, th flag i not moving. Mind i moving. Trnc J. Nlon A impl foundational principl for quantum mchanic i propod lading naturally to a compl probability dnity prd a a um of path intgral. Th action i thn drivd in it rlativitically-invariant form with th Lagrangian following from th principl intad of bing impod a an tra hypothi. Howvr, th um i rtrictd to phyical path on which th particl ha ral vctor momntum. Thi rtriction vidntly conflict with th uual drivation of Schrodingr' quation and from it conrvation of nrgy and probability a dfind by Born. To if th rtrictd path intgral um lad to nibl prdiction, th doubl-lit primnt i analyzd uing a ingl-quantum tranfr approimation. Th rult i found to b in gnral agrmnt with primnt providd w add a natural maurmnt hypothi. Howvr, it do prdict that nrgy i not actly conrvd. Th mot probabl chang in nrgy (nar th poition of th firt minimum at th crn i a lo that i proportional to th quar of Planck' contant whn prd in trm of th lit gomtry and th ma of th particl. Introduction Whn I firt hard about Hinbrg' uncrtainty principl, I naivly uppod that th tatitical natur of th prdiction of quantum mchanic would mrg from th unprdictability of individual chang of quanta btwn fild and matrial particl. Howvr, th dvlopmnt didn't tak thi path. What nud wa bt charactrizd by Richard Fynman whn h aid [] "w cannot mak th mytry go away w will jut tll you how it work." Aftr thinking about it for a long tim, I may hav found a impl way forward along th path I had originally anticipatd. Th firt part of th foundational principal (notably miing from convntional quantum thory [] that I want to propo i: a Particl and fild intract dicrtly through quantizd chang of momntum that occur at prdictabl rat but at unprdictabl tim. I hould add that I think thi i obviou bcau on could only dicovr th timing of a firt chang with additional intraction that would compound th ituation.. Building on th Foundation Th forc acting on a particl i th ngativ of th gradint of th potntial nrgy. Whn th potntial, ay V(, i prd a a Fourir intgral, th intgrand diffr from that of th potntial by a factor of ik, Zn Flh, Zn Bon - A Collction of Zn and pr-zn Writing compild by Paul Rp, Doublday, Nw York, 96, p. 4.
F V( ( i d k V 3 ( / k ( k ik On th othr hand, ħk i th momntum pr quanta changd with th fild. What i lft, formally at lat, hould b th rat of chang of uch quanta in d 3 k, and intgrating thi rat ovr k-pac thrfor uggt that th total rat of chang at all wav vctor i i(/ħv(. Formally applying Poion tatitic and ponntiating th ngativ of th pctd numbr of chang in a tim dt thn giv a "probability" Pa ( i / V( dt that no chang btwn a particl at and th fild occur in tim dt. (Th ubcript jut indicat that w ar not don dfining th total probability yt. Now, w till nd a trm that rprnt th ffct of inrtia. If th potntial nrgy i a maur of th avrag rat at which quanta ar changd with th fild, th mchanical nrgy could b rlatd to intraction with whatvr it i that giv particl thir ma and inrtia. Adding th total mchanical nrgy T to th potntial V giv: Pb i(/ ( V( + T( p dt for th probability of no chang in tim dt. Hr p i th momntum of th particl and T(p i th total (kintic + rt mchanical nrgy. Now conidr a particl tarting from location at tim t with momntum p. Th probability dnity for finding it till on it initial trajctory at om latr tim t will b: i(/ dt'( V( ' + T( p 3 δ ( ( ρc(, t ;, t, v v t t bcau multiplying th probabiliti for ach incrmnt of tim rult in adding th diffrntial trm in th ponnt. Thy add up to th path intgral of th um of th potntial and mchanical nrgi. Of cour th initial momntum and poition can t b known imultanouly bcau intrumnt that might b ud to dtrmin thm ar alo unprdictabl at th quantum lvl. Nvrthl, th claical mtaphyical principl that a particl ha a uniqu poition and momntum i not ncarily abandond. Owing to our unavoidabl ignoranc, w will jut hav to avrag ovr poition or momntum or om combination throf in a practical ituation. Continuing with th formal application of Poion tatitic uggt that th probability of chang of a quantum with wav vctor k in d 3 k during a tim dt i: P i( / dtd kv( k k + + p d 3 i i(/ ( V( T( dt Aftr picking up momntum ħk, th particl will b on a nw trajctory until th nt quantum i changd. Th amplitud for arriving at om final poition at tim t can b prd a
an intgral along th initial trajctory of a product of thr trm. Th firt trm i th probability for not intracting along th initial lg, which i givn by P c if w ay th firt quantum i changd at t. Th cond trm i th amplitud for th firt chang, which modifi th trajctory th particl will follow to th cond chang: p p + k, + ( t t v Th third trm i th amplitud for arriving at th final poition at tim t with tarting poition hiftd to and t. Thi raoning uggt an intgral quation: 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 hould b notd that d 3 kv(k ha th dimnion of nrgy, and th ffctiv intgration ovr th pac-tim coordinat i rtrictd by rlativitic cauality. In othr word, th intrval btwn,t and,t and btwn,t and,t mut b tim-lik. W can rwrit th pha factor that com from th Fourir tranform accuratly a: ik (/ i ( p p (/ i d p' ' bcau th incrmnt in momntum i concntratd at th nd of th intrval whr. Thi trm combin with th path intgral of th nrgy to produc a factor of th form: A i(/ dt'( V( ' + T( p' + dp' ' which appar to b cloly rlatd to th Fynman path intgral [3]. In fact, if w intgrat th third trm in th intgrand by part onc, w find i(/ dt V( i(/ ( p p T( A ( ' + p' p' v' i(/ dt' L( ', p' i(/ ( p p contain Fynman path intgral of th Lagrangian L(, p multiplid by i(/ ( p p 3
Now, th vlocity btwn t and t i v, and ħk p -p, o w can rarrang trm to pr th quation in a potntially implr 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 ( ( (/ ( (, ;,, Thi can b viwd a an intgral quation for th probability dnity (for th particl to arriv at and t multiplid by a pha factor aociatd with th (hypothtical initial momntum and poition. W ar lft with th path intgral of th Lagrangian and th Fourir tranform of th potntial to dal with in actual calculation. A in Fynman formulation, contribution from narby path hould add in pha whn th Lagrangian i clo to an trmum and othrwi cancl. Howvr, only phyical path ar gnratd by conidring quantizd tranfr of momntum. Fynman' formulation includ all path in coordinat pac ubjct along which tim do not rgr. Thi man that th particl i rquird to mov fatr than th pd of light on om of th path. On th othr hand, th prion for th Lagrangian that rult from our hypoth L V( + T( p p v whil corrct claically, bcau p. v i qual to twic th kintic nrgy, alo giv th corrct quation of motion whn th nrgy and momntum ar corrctd according to th pcial thory of rlativity. In fact, th intgral of th Lagrangian appar in it Lorntz invariant form ( ( p ( p v T + V dt p d µ It may b that th incluion of unphyical path in Fynman' formulation, ing that it i thn compatibl with Schrodingr' quation, i jut a vry convnint approimation. For ampl in olving th doubl-lit primnt, it i air to u olution of Schrodingr' quation than to comput th path intgral plicitly. Howvr, matching up olution of Schrodingr' quation in diffrnt rgion forc th nrgy with which th particl rach th crn to b actly th am a it had approaching th barrir. It m that act conrvation of nrgy ultimatly follow from th incluion of unphyical path in th path-intgral formulation, and calculating th amplitud a a um rtrictd to phyical path could giv a lightly diffrnt anwr. At lat on mor hypothi i ncary in ordr to b abl to prdict th rult of maurmnt. Sinc our compl probability dnity i rlatd to Fynman' path intgral, minu th unphyical path, th implt choic i to procd a in tandard quantum mchanic. That, i th maurmnt probability dnity hould b proportional to th quar of our compl probability dnity. Howvr, th intgral of thi dnity ovr th patial coordinat might not b contant in tim if Schrodingr' quation i not atifid. Thrfor it m to b ncary 4 µ
to conidr tranint rathr than tady-tat vnt and intgrat ovr tim to gt th maurmnt dnity: b. Th probability dnity of a maurmnt i proportional to th intgral ovr tim of th quar of th abolut magnitud of th compl probability dnity for th ytm bing maurd to arriv at th point of th maurmnt. Thi procdur m to b raonabl providd w conidr vnt that ar iolatd in tim or ufficintly infrqunt o that intrfrnc ffct btwn ucciv particl in th rcording mdia bcom ngligibl. Thi, in turn, may impo an primntal uppr limit on th timcal for iolatd vnt in practical ituation. Sinc w hav alrady drivd th Lagrangian, w can ay that th claical quation of motion follow from our quantum hypothi vn if it dpart in om rpct from th prdiction of convntional quantum mchanic. That thr i om dpartur m clar from th fact w do not includ unphyical path and do not pct Schrodingr' quation to hold actly. To clarify whthr or not our quantum principl can agr with primnt, it m appropriat nt to invtigat a familiar problm, th doubl-lit primnt. W do find th nrgy i not actly conrvd. In non-rlativitic ca, th chang in nrgy i ngativ and on th ordr of th kintic nrgy that th particl would hav, aftr cattring, du to it motion paralll to th plan of th lit multiplid by a dimnionl function of th lit gomtry. (It can b gratr for narrow lit. A th ratio of th vlocity paralll to th plan of th lit dividd by th prpndicular componnt i qual to th cattring angl, th ratio of th nrgi i proportional to th quar of th cattring angl, and th mall prcntag chang in nrgy might b hard to dtct.. Quantum Mytry Diffraction by Slit Fynman ud th doubl-lit primnt a an ampl and dicud it in grat dtail bcau, h aid, "it contain th only mytry." Th only mytry, of cour, i that particl appar to bhav lik wav in thi primnt. Thi wav-lik bhavior appar a intrfrnc fring in th ditribution of particl that pa through th lit v. th angl btwn th incoming and outgoing momntum of th particl. Fynman ud th doubl-lit primnt a a pdagogical tool. Sinc th tim of hi lctur, th pctd intrfrnc bhavior ha bn confirmd dirctly in lgant primnt [4]. Eact olution to th quation dvlopd abov ar not known yt. Whn th initial path from along v lad to a colliion with th crn, th panion may convrg lowly, if at all, but th rult mut b mall if i on th othr id of th crn from. Whn thr i no colliion, w may b abl to conidr only th lowt-ordr trm. Th compl probability dnity that a particl on a trajctory paing through, t will arriv at,t in thi approimation i 5
ρ (, t;, t, v tma (, i( / dt dk dk dk V ( k δ ( v ( t t tmin (, i(/ T( tt i(/ ( p p i(/ T( tt 3 y z ( t t ( t t + v, p p + k, + v providd i not on th original path. For ach lit, th targt poition i viibl from th poition of th particl btwn th limit in th intgration ovr t. Howvr, if th initial trajctory actually pa through th lit, th uppr limit i dtrmind by rlativitic cauality intad. Th intgration ovr t includ tim whn th particl can cattr bfor paing through on of th lit on it way to th obrvation point. That i, th lowr limit t min i gnrally l than t. Th uppr limit occur whn th targt i no longr viibl from th poition of th particl. Th window do not clo, howvr, if th initial trajctory pa through on of th lit. In that ca th uppr limit i dtrmind by rlativitic cauality intad. For implicity, lt aum th particl initially approach th plan of th lit along th normal and call thi dirction th z-ai. Th potntial i actually indpndnt of poition along th ai paralll to th lit, which w ll call th y-ai. Th dpndnc of th Fourir tranform of th potntial on th corrponding componnt of th wav vctor will thrfor concntratd into a dlta function at k y. Thu th outgoing momntum will rmain in th original plan prpndicular to th dirction of th lit. Undr th aumption, th firt-ordr trm abov implifi to ρ (, t;, t, v tma (, i( / h dt dk dk V( k, k δ ( v ( tt p p + k k y tmin (, i(/ T( tt i(/ ( pp i(/ T( tt 3 z z Bcau of th dlta function, it i convnint to convrt th doubl intgration ovr dk dk y into quivalnt intgration ovr dv dv z. Th Jacobian for thi chang of variabl i 6
( k, k k v k 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 dlta function tak car of th intgration laving ρ (, t;, t, v (,, v v tma t 3 i(/ T( tt V( k, kz i(/ ( pp i(/ T( tt δ min (,, ( tt ( v t t im ( / π dt ( y y p p + k + v ( tt whr th uppr limit of th intgration ovr t can b t by rlativitic cauality. Thi will b th ca if th initial trajctory pa through on of th opning. Lt alo aum th plan from which th lit ha bn cut i vry thin and th z-dpndnc of th potntial contain a dlta function at z. Thn th Fourir tranform of th potntial i actually indpndnt of k z o th probability dnity can b writtn ρ (, t;, t, v (,, tma t 3 i(/ T( tt V( k i(/ ( pp i(/ T( tt δ min (,, ( tt ( v t t im ( / π dt ( y y p p + k + v ( tt. Equation for Stationary Pha Nt, w anticipat that contribution to th intgral will b mall cpt whn th pha of th intgrand i tationary. Th ait way to whn that happn i to not that + v ( t t 7
Thi allow th pha in th intgrand (in unit of ħ to b prd in th particularly impl 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 Thi pha i tationary with rpct to th tim of th intraction whn Th drivativ of th Lagrangian i givn by dφ dp dl + ( L L + ( t t dt dt dt dl T dv v dt c dt and th drivativ of th momntum i rlatd to th vlocity and it rat of chang 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 tationary pha can 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 aftr th intraction can b convnintly writtn a 8
v v z v ( t t a 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 hav introducd th vctor a pointing to th final poition from th poition +v (tt, which i whr th particl would b at tim t if it continud on it original cour without intracting. (S Figur. for th rlation btwn th vctor. In turn, th rat of chang of th componnt of v ar traightforwardly calculatd to b dv ( tt v dt t t a ( t t ( Thu th rat of chang of v with rpct to t i contant in dirction a, but it divrg in magnitud a t approach t. W can u thi to rwrit th condition for th pha to b tationary a a v v ( L L (( ctt ( v c c T v( t t + + If th intraction occur vry lat, v can bcom apprciabl compard to th pd of light c. Othrwi, c(t-t will b larg compard to othr quantiti in th quation with th dimnion of lngth. Thi man that th firt trm of th quation will b mall, and th olution will b found nar L L. Thr will b a olution (for fid t bcau th cond trm tart out ngativ (if w includ tim t bfor t, pa through zro whn nrgy i conrvd actly, and thn bcom poitiv for latr valu of t. Th condition for tationary pha can b rwrittn a / / a v v v t t (( ( / v v v ( + c c c c t t v c c c Squaring thi, and implifying omwhat, lad to 9
v v a +ε a +ε + ( c c c tt c ( tt ( v c ( tt c ( tt a v v a +ε a ( +ε + + ct ( t c c ct ( t ct ( t ( v ct ( t ct ( t a v v +ε ( +ε a ( +ε + + ct ( t c c ct ( t (v c( tt c( tt c( tt a b+ε +ε a +ε ct ( t ct ( t ( v ct ( t ct ( t ct ( t whr w introduc th dfinition v v ε ( v c c v v v + ( v c c c b ( v+ v ( t t + a + v( t t + a + v( t t + for th vctor b, which i anothr vctor that i contant in dirction, and ε, which i mall compard to b in th non-rlativitic limit. Providd t-t i not infinit, th tationary-pha condition bcom quivalnt to a ( +ε b+ε v c( t- t a ( Th right-hand id i alway poitiv, but it will rmain vry mall a long a c(t-t i larg. Conquntly a mut b narly orthogonal to b+ε. Evn if a bcom vry mall, it will till hav to b narly orthogonal to b bcau th right-hand id cal lik a. If, to form th nonrlativitic limit, w nglct both a. ε and th right-hand id altogthr, w hav a b ( v( tt + v( t t + a + + ( v ( t t ( v ( t t ( t t + v
At thi lvl of approimation, th pha i tationary at a particular valu of t and i indpndnt of t. Thi uggt that th path intgral of th Lagrangian may not alway vary a rapidly from it tationary valu a i uually aumd.. Solution Auming Enrgy i Conrvd Eactly To gt a bttr approimation, it will b ncary to handl crtain mall quantiti dlicatly. Anticipating that nrgy will b approimatly conrvd, w firt not that v a v + c c( t t c a t t v + c ( a+ v ( ( ct ( t a ( b v ( ct ( t c + Thi how that quar of th vlocity, and hnc th nrgy, will b changd vry littl by th intraction, providd c(t-t i larg compard to any of th quantiti in th problm with th dimnion of lngth. W can alo that nrgy would b conrvd actly if a wr orthogonal to th vctor b-, which giv a valu of t -t Thi i quivalnt to a ( b t t a v t t tt ( tt a ( b tt a v a a v Thi provid on rlation btwn th rmaining two fr paramtr t and t. Now, if a. b i qual to a., thn it follow that a. (b+ε i qual to a. ( +ε. Uing thi in th quation for tationary pha giv ( a +ε ct ( t ( v /
if it i prmiibl to divid by a. ( +ε. Howvr, thi rult i vidntly incompatibl with non-rlativitic valu of v, o it follow that both a. (b+ε and a. ( +ε mut b zro. Thn, Figur. Dfinition of Variabl in Doubl Slit Eprimnt aftr multiplying through by (-v /c [c(t-t ], th quation for tationary pha implifi, in th non-rlativitic limit whn nrgy i conrvd actly, to ( a b+ε v v ( ct t ( ct t a ( v c( t- t ( ( - a ( - a b c c v v + a a+ ct ( - t a+ ct ( - t c c v v v ( ct (- t a b a + c c c v v v +( ct (-t a a b+ a a+ a c c c a a b+ a a Subtituting th valu for t-t that corrpond to act conrvation of nrgy, w ar ld to th following quation
a v v v a b a a v c c c a v v v a a b+ a a a a v c c c 4 + + + + a On dividing through by th known rult for[c(t-t ], thi bcom v v v a b+ a c c c a v v v v + a a b+ a a a a + c c c 4 a v + a a v v v a v 4 a b a + a + a a ba a c c c a v v v a v a b a + { + a a} c c c a { } rlating th componnt of a whn nrgy i conrvd actly. Sinc w aumd i a fr paramtr, thi condition fi a. v, or quivalntly v (t-t, a a function of. In othr word, it dfin a curv of ndpoint of trajctori for which th pha i tationary and nrgy i conrvd. Not that whn a i narly prpndicular to v, th rulting valu of a. b bcom ( a vˆ v ( tt v a b a c c v / v a + / v c ( a vˆ v ( t t ( a v c whr th cond rult follow if it can b aumd that (a. v /c i ngligibl compard to (- v /c a. Howvr th act rult i quivalnt to a 4 th ordr quation for t-t or quivalntly a. v /v in trm of a (or quivalntly whr v /c and th componnt of ar givn paramtr. ( a v ( b v v v c ˆ ˆ / 3
In gnral, w hould pct a. v to b ngativ and trmly mall compard to b. v whn nrgy i conrvd. Furthrmor, a. b will alo b ngativ with magnitud on th ordr of v /c tim -. Subtituting into th olution for t -t, w find a ( b v( t t a vˆ v c a / / v( t t v c a vˆ / / v ( t t bvˆ v c b v c v c v c + / / which how that a bcom ngativ nough, th intraction can occur bfor th particl rach th plan of th lit and till conrv nrgy. In that ca, of cour, th particl can only rach th targt if it intrct th plan through on of th lit..3 Uniqun of Eact Enrgy Conrvation Aftr multiplying through by -v /c, th quation for tationary pha can b rwrittn ( ( ct ( t v a b a ( b+ε a ( b ( a b+ε c c(- t t Hr w notic that cro product of a. (b+ε and a. (b- occur on both id of th quation, and by collcting th trm on th right-hand id, w ar lft with v a ( b+ε + a ( b ( a b+ε c c(- t t W hav now tranformd th quation for tationary pha into an quivalnt on whr th variabl factor -v /c do not occur (cpt in th dfinition of ε. Nt, not that inc b + v ( tt a b th tranformd quation can b rwrittn uch that ε i aociatd with v a a ( +ε v ( a +ε + a c c( t- t c which giv u a quadratic quation for a. (+ ε/c(t-t 4
v v v v + a ct ( - t + a + a a ( +ε a ( + ε ct (- t ct (- t c c c c Th appropriatly mall olution i a ( +ε v v v v v v a + ct ( - t a + ct ( - t a ct (- t c c c c a c c Howvr, thi i quivalnt to / ( b+ε a v v v v v v + a + ct ( - t a + ct ( - t a (- (- a a c t t c t t c c c c c c and, noting that a. (b+ε i rquird to b poitiv, w that th quation for tationary pha actually ha no olution whn th inquality a v v v v v v >a + ct ( - t a + ct ( - t a a (- ct t c c c c c c hold. Iolating th radical on th right-hand id of th inquality and thn quaring giv a v v v v a + ct ( - t >a a a (- ct t (- ct t c c c c By multiplying through by[c(t-t ], w thi i quivalnt to v 4 [ ct t ] a a [ ct t ] a v a (- + (- + > c c 4 Thi inquality obviouly hold for larg c(t-t but would b fal btwn th zro of th quadratic form on th lft hand id. Howvr, th two zro occur at th am valu a ct (- t a v corrponding to act nrgy conrvation, which i thrfor th only olution. / / 5
.4 Stationary Pha Condition with Tranvr Momntum a th Indpndnt Variabl Eact olution to th quation for tationary pha ar not known yt whn nrgy i not conrvd. W hav trid an approimation mthod bad on mall dpartur from nrgy conrvation, but unfortunatly th pha do not ncarily vary rapidly with mall dpartur from th valu of t at which th pha i thn found to b tationary. An altrnat approach will b followd in which th indpndnt variabl i changd from t to k. Subtracting trm in th pha that ar indpndnt of t-t, w ar lft to conidr Th firt trm can b rwrittn φ L ( t t + p p + ( L L ( tt T a v + c c t t c a T v a c c p ( k + It will b uful to u k a th indpndnt variabl intad of t. To carry out thi program, w can olv k p T v c mca / { v } c( tt mca v ( v c( t t a c( t t a c for c(t-t in trm of k. Squaring th quation abov lad to a quadratic quation in c(t-t for which th poibl olution ar / ( / ( ( a v a v mca v c c t t ± a / + + ( v v k It clar that th uppr (+ ign i th on w nd if th intraction occur bfor th obrvation. Our fundamntal phyical quantity i a pha hift pribl a a product of an 6 / /
nrgy and a tim intrval. Howvr, w may hav th option, firt rcognizd by Dirac, of chooing th minu ign and intrprting th olution moving backward in tim a an antiparticl. For th problm at hand, though, w can u T c k ( t c t a k / ( a v / a v c c mca + / / a + + a ( / ( / v k v c v c ( a v ka v mc k + / + a + a ( / ( / v mca v c v c / / to valuat th cond trm in p.. Adding it all up, w find a T v φ L ( t t + p k + + ( L L ( tt a c c a T v T k + + mc( v c( tt ( v / / a c c mc a T / v mca T ( / a c c k mc k + + mc v c mca k mc + mc a + k mca c Sparating out th radical trm in th prion abov for T /c, φ L ( t t + p v mca k ( v / T c ( v ( a v mca a v k mc mc + mc / a + k ( v ( / mca v c v / mca T ka v + ( v c k c a ( v Finally, including th radical trm plicitly,and chooing th +ign to gt poitiv total nrgy, th pha can b writtn 7
φ L ( t t + p ( ( v / c / ( v ( a v mca a v k mc mc + mc a / + k mca v ( a v mc v / mca k + / ( v + a + ( v c k v mca a v k v a v mc + mc a + v / mca v c ( ( ( / mc v / mca ( a v / c k + / ( v + a + ( v c k v mca / Thi prion i wll bhavd at low momntum tranfr (ħk <<mca /a, and to cond ordr in k giv φ L( t t + p ( a v ( v ( a v ( a v mc + mc + a + ( v mc v ( a v k + a / + ( v c v mca ( a v ( v a b+ ( b v ( a v k mc + mc / ( v ( / mc v c a v ˆ mc a v k / + + / ( v c ( v a mc k a / v v mca On th othr hand, at high momntum tranfr (ħk >>mca /a, th pha can b approimatd by a powr ri in mc/ħk 8
φ L( t t + p ( a v k ( v ( a v mc + mc / a + ( v mca v / mc v / ˆ mca k a v mc + / ( v / ( v c k mc + + ( v a k ( v ( a v ˆ k mc v a v k mc a + + + / / v mca ( / c v c ( v a mc ( a v a vˆ + mc mca... / + / + ( v ( v a In thi limit, th pha ha a componnt that i linar in k with a larg cofficint on th ordr of. v /c..5 Intgration ovr Tranvr Momntum Intad of Intraction Tim W will now carry through with th chang of variabl in th prion for th compl probability dnity in th ingl quantum chang approimation ρ (, t;, t, v (, tma iφ 3 V( k δ min(, ( tt ( v t im ( / π ( y y dt from t to k. W can do thi by diffrntiating p, which i a function of t-t and at th am tim qual to ħk / 9
dp T dv T v dv + v dt c dt c ( v c cdt T dv T a a v v + + c dt c v c c t t t t ( / c( t t ( T a a v + + c v c c t t v ( / ( c( t t t t T v c ( v ( a v v c t t tt Th combination of factor in th intgral that w nd thrfor bcom ( ( / dt cdk tt ( v a v c T a v v c( t t dk c ( t t / ma ( v / / ( v ( v c( t t a v dk mca ma ( v / / ( ( a v mca v k a + + v k dk / / ( a v k ( v ( v ma + a + v mca / Thu aftr th chang of variabl, th compl probability dnity bcom ρ (, t;, t, v (, v ( v k ma iφ 3 V( k δ / k min(, / ( a v k ( v ( v ma + a + v mca im ( / π ( yy dk Thi can b uful bcau th limit of th intgration ovr k dpnd on whthr th initial
trajctory pa through th lit in qution and ar othrwi almot fid valu (for ach lit..5. Intgration ovr Tranvr Momntum on Blockd Trajctori Rfrring to Figur, w that th lowr limit of k (a a function of corrpond to th arlit intraction tim t min whr. ma v ( t t + a vˆ v ( t t + a vˆ min Thi rult in th particl intrcting th plan at th top ( ma of lit. Solving for v (t-t min, v ( t t v t t + a vˆ ( ma min ma ma ( v ( t t a vˆ + a v ma v a v ma Auming th initial trajctory do not pa through th lit, a imilar rlation it btwn th maimum valu of t and th poition of th bottom of th lit ˆ ˆ ˆ Figur. Dfinition of Minimum and Maimum Intraction Tim
v ( t t v t t + a vˆ ( ( a vˆ ma min v a v min It thrfor turn out that whn a. v i mall compard to. v, ( w hall that it i vry mall th -componnt of th vlocity at th two limit ar mall and approimatly ˆ ˆ v ma v min indpndnt of. ( v ( min v vˆ a vˆ vˆ ˆ a v min min ( v vˆ min ( ma v vˆ a vˆ ma ( v ma vˆ Th vlociti ar till lightly diffrnt for ach lit. In th am approimation, th componnt of th vlocity along th initial dirction rmain ntially qual to v. Thi man that th momntum tranfr i mall and narly contant at k mv ( + / mv mv min ma vˆ ˆ v It will b noticd that th diffrnc in man momntum for th two lit i mall in proportion to th ditanc from th ai of ymmtry to lit poition compard to th targt poition k k whr k -bar tand for th avrag of th man momnta of th two valu of k. Th width of th rang of th intgration will b
( δ k m v v ma min mv vˆ w mv vˆ ma min and it magnitud compard to th man momntum tranfr δ k w k will alo b mall if th lit gomtry i comprd in ordr to mak th firt minimum occur at a largr valu of whr it hould b air to obrv. Of cour, th limiting and man vlociti ar lightly diffrnt for ach lit, o th compl probability dnity bcom approimatly ρ (, t;, t, v av v k ma(, iφ i( / π δ( yy V( k dk / k min(, / ( a v k ( v ( v a + a + v mca In a typical ca, th man valu of momntum tranfr for th two lit ar not much diffrnt, o w gt th pctd form that i proportional to th Fourir tranform at th wav vctor that corrpond to cattring in th vicinity of th lit. To cond ordr in ħk /mc and a. v /a, but cluding trm of cond ordr in both at th am tim, th pha can b writtn ( a v φl ( t t + p mc + k ξ + σ k whr w hav introducd th dfinition ( v ( ( v { } / ξ ξ σ k a v mc + + / σ σ 3
ξ σ ( v a b+ ( b v ( a v ( v a v / / ( v c mc Not that, to th tnt that a. v i mall compard to a v, th cofficint of th quadratic trm in k i approimatly indpndnt of. Th Th pha i tationary cntrd at momntum tranfr / k mc tationary ξ mc σ ( v c / / ξ v Th linar trm in th pha bcom ignificant whn k ξ i of th ordr of unity or largr. In a non-rlativitic primnt, th quadratic trm in th pha i likly to b mall on blockd trajctori, a can b n by rarranging ( σ k v mv ( v v mv / mc vˆ / ˆ c v ( v c / Th cond factor would b about ±π/4 if corrpond to th firt minimum at th targt. Suppoing w hav chon to b about tim -bar, in ordr to mak th firt minimum mor aily obrvabl, th firt factor will b l than (v /c and would b mall if v /c i l than about -. Th pha will vary rapidly at valu of momntum tranfr whr rlativitic ffct ar ignificant, that i if ħk /mca i not ngligibl compard to unity. Howvr, th intgrand rmain boundd bcau th Fourir tranform of th potntial contribut a factor of k in th dnominator 4
V( k kv( k a / ( a v / k mc c k v ( / a a v c + + v mca mc kvk ( a v mc ( which along with th factor that wa alrady thr, combin with th Lorntz factor (-v /c to giv v /c which approach unity in thi limit. Thu for th blockd trajctori, th compl probability dnity bcom approimatly ρ (, t;, t, v av v k ma(, V( k iφ i( / π δ( yy dk 3/ ( v a k min(, ( a v ( i/ L ( tt p + mc v / c k ik a ξ + / ( v mc / ( v k m (, V( k i( / π δ( yy dk 3/ v a k, ( min( Not that th contribution from trajctori paing though ach lit ar paratly proportional to V(k -bar, and ach how th minimum that i attributd to dtructiv intrfrnc in convntional quantum mchanic. In typical ca, th tranform of th potntial i almot th am for trajctori paing through th two lit, o thir contribution do not ncarily cancl or intrfr. Noting that whn th quadratic trm i ngligibl ovr th rtrictd rang of momntum tranfr on blockd trajctori, w can valuat th intgral ovr dk v / c k kma(, ik ξ + / k ma(, ( / mc v c ik { ξ+ σ k } dk dk kmin(, kmin(, k ma(, ik ξ dk k min(, ik (, ξ ik (, ξ δ k ma min ik ξ iξ in δkξ / δk ξ / 5
.5. Intgration ovr Tranvr Momntum on Unblockd Trajctori If, on th othr hand, th initial trajctory do pa through a lit, th uppr limit of th intgration ovr k bcom infinit. Rlvant to that limit, th Fourir tranform of th potntial + V( k dv( π V δ ( π ik ik V k d lit L/ L/+ W ik V Vδ ( k + π ik L/W L/ V Vδ k + k L + W k L { } ( in ( / in ( / kπ contribut anothr factor of k in th dnominator. W hav aumd that th dlta function i outid th rang of th intgration ovr k, o w hav ρ(, t;, t, v V { in k ( L/ + W in k ( L/ } im yy dk iφ 3 ( / π δ( / k min(, / ( a v k π ma( v ( v k + a + v mca At high momntum tranfr, th dnominator of th intgrand cal lik k (-v /c (mv /ħ, which approach a contant valu bcau v i boundd by c. Alo in thi limit, th pha i again approimatly linar in k, but it cofficint i on th ordr of.v /c, which i larg in magnitud compard to L/+W and L/ in th ca of primary intrt. Thrfor, th intgrand will ocillat rapidly and contribut littl to th compl probability dnity. In typical ca, ħk min /mc i on th ordr of ( k mc v min ma vˆ c auming that a. v i mall compard to. v. On th othr hand, bcau w ar intrtd in th firt null in th diffraction pattrn, 6
L+ W W L+ W W in k + in k in k( L/ + W in k( L/ k k L+ W W cok ink k it follow that k min (L+W/ will b of th ordr of unity, or mor pcifically π/. Thrfor, whn th rlativitic corrction bcom ignificant, k will b many ordr of magnitud largr than k min, and th intgrand will b many ordr of magnitud mallr,. Thrfor, ocillation at a mall but finit amplitud at high momntum tranfr can b ignord in a firt approimation and w can conidr ρ (, t;, t, v V i( / π ( y y dk δ π a v c 3/ ( / k min(, ( a v ( i/ L( tt p + mc / ( v V dk δ 3/ a v π k k min, i(/ π ( y y in k ( ( { in ( / + in ( / } k L W k L { ( ( } { ik } ξ+ ο k L/+ W in k L/ It i uful to tnd th lowr limit of th intgration to and thn conidr th compnating intgral ovr th finit limit paratly ρ (, t;, t, v ( a v ( i/ L( t t p mc dk + / iο k ( { in k ( / in ( / } / L W k L v c + V k + iin kξ δ 3/ k min(, a ( v π dk L+ W W ik { ξ+ ο k} cok ink k i( / π ( y y k iφ co k ξ ik ξ+ ο k in ( / in ( / ( / k L + W k L a v c k ( i/ L ( tt p + mc / ( v V dk 3/ ( ( a ( v π k k min(, dk L W W ik { ξ+ ο k + } cok ink k { } { } { }( i(/ π δ( y y + i in k L/+ W in k L/ ink ξ which i poibl bcau th intgrand i wll bhavd at k. Th ral part of th infinit intgral rproduc th potntial valuatd at ξ with om lo of high patial frqunci du to th quadratic trm in th pha. If thi lo can b nglctd, that i σ can b t to zro, w thn hav 7 dk iο k
L L { ( ( } { } W ξ dk ik in ξ+ ο k π < < k L/ W in k L/ + k L L < ξ < + W In othr word, thi part of th rult i a non-zro contant whn ξ, which ha th dimnion of lngth, i in on of th lit. If w t σ qual to firt, th rmaining imaginary part of th infinit intgral i i k L W k L k ( L/ + W + ξ( L/ξ ( + ξ( + ξ dk i { in ( / + in ( / }( in ξ ln k 4 L/ W L/ according to Gradhtyn and Ryzhik [5] 3.74 # (pag 44. Thi rult ha logarithmic ingulariti whr ξ coincid with th dg of th lit, but it bcom rapidly mallr a ξ mov away from th lit. For th tim bing, it will b aumd that thi form giv corrct rult aftr th intgration ovr d. Finally, if th quadratic trm in th pha can b nglctd in th infinit intgral ovr dk, it will b raonabl to nglct it in th intgral from to k min a wll ρ (, t;, t, v L L W ξ π < < L L < ξ < + W ( a v ( i/ L( tt p + mc / ( v V i ( L/ + W + ξ( L/ξ i( / π δ( yy ln 3/ a ( 4 ( / ( / v π L W ξ L ξ + + k min(, dk ik cok L W in k W + ξ k.6 Intgrating Ovr th Bam So far, w hav calculatd th compl probability dnity rulting from a ingl particl ntring with vlocity v along th z-ai on a trajctory paing through om fid. Of cour, w don t know how to prpar that initial tat. Mot likly, w would hav a ditant point ourc, and th diplacmnt of th initial trajctory from th z-ai would b unknown. Bcau of th tranlation invarianc of th boundary condition along th y-ai, intgrating th probability dnity ovr y will rplac th dlta function δ(y-y by N y which i th dnity of particl in th y-dirction. Likwi, adding up th contribution in th -dirction ntail multiplying by a factor of N a wll a intgrating ovr d. Th final rult, of cour i thn indpndnt of. 8
.6. Intgrating Ovr th Bam on Blockd Trajctori For th abov raon, for blockd trajctori, th avragd contribution bcom ρ (,; tt, v ( a v ( i/ L( tt p + mc / ( v min d ik k ξ δ y 3/ δ v a δk i( / π N N V( k k ( in ξ / ξ / Th intgral ovr d can b put in a mor tandard form by changing th variabl of intgration to ξ,, which giv ( bv ˆ ( av ˆ a d dξ + ( v a a ± ( ξ + / ( bv ˆ ( av ˆ ( v c / d a It alo follow from th dfinition of ξ that th pha trm k ξ of th intgrand incra monotonically with ovr th rang (-, in which a i poitiv if ( b vˆ ( a vˆ ( vˆ v ( tt v c v c > / / ( Othrwi, (that i if a. v < ξ pa through a maimum valu ξ ma at ( p ξ ma p at th valu of p (for tationary pha givn by ( bv ˆ ( av ˆ / p v c / ( vˆ v ( tt v / Figur 3 how ampl of k ξ plottd v / at diffrnt valu of p / auming th incoming particl i non-rlativitic and -bar/., v /. v -5 and mv /ħ.( 7. 9
Solving for a. v, w find ( a vˆ vˆ( a vˆ ( v c ( p + / > (( ( v c ( p ( v ( p / vˆ a vˆ > vˆ + vˆ + / Roughly paking, th minimum valu of - a. v /v i a mall compard to - p a - p i to. v /v, It will b notd that act conrvation of nrgy on th initial trajctory dfind by p can b in thi rang. Undr th aumption that (a. v /c i ngligibl compard to (-v /c a, nrgy i conrvd if v p < v Of cour, th initial trajctori that contribut th mot to th compl probability dnity alo corrpond to mall valu of ξ ma p. Upon th chang of variabl, th trm that thn appar in th dnominator, intad of a, upon intgrating ovr dξ intad of d can b valuatd by olving th dfinition of ξ a a quadratic quation for a, ( bv ˆ ( av ˆ a ( ξ ± ( ξ + / ( v c / giving ( ( ( ( a bv ˆ av ˆ bv ˆ av ˆ + ± ( ξ + ( v a v ( / For poitiv valu of a th poitiv ign i rquird whn a. v i poitiv but whn a. v i ngativ, th ngativ ign i ndd in th rang p < < min o d p dξ a p ( bv ˆ( av ˆ ( ξ + ( v / and th nt rult of changing th variabl of intgration i 3
Figur 3 Non-rlativitic doubl lit primnt with -bar/., v /. v -5 and mv /ħ.( 7. ρ (,; tt, v ( a v ( i/ L( tt p + mc / ( v ( / π y ( 3/ δ i N N V k k ξ ( ( v min p dξ ik in / ξ δk ξ / ( ( / p bvˆ ˆ av δk ξ ( ξ + ( v If p i in th rang of th intgration th intgral ovr dξ, it will b th maimum valu rachd by ξ, and th intgral can b conidrd in two part 3
ξ ( min p dξ ik in / ξ δk ξ / ( ( / p bv ˆ ˆ av δk ξ ( ξ + ( v p p dξ ik in / ξ δk ξ dξ / + ( ( / bvˆ ˆ av δk ξ ξ ( min bv ˆ ˆ av ( ξ + ( ξ + v v ( ( ( ( / ik ξ in δkξ / δkξ / If, in addition, k -bar i much largr than δk (making th firt minimum at th targt mor aily obrvabl th pha trm k -bar ξ will bgin to ocillat rapidly bfor δk ξ/ bcom ignificantly diffrnt from δk p /. In uch ca, th intgral ovr dξ bcom approimatly ξ ( min p dξ ik in / ξ δk ξ / ( ( / p bv ˆ ˆ av δk ξ ( ξ + ( v p ik in / ξ p ik ξ δ k p dξ dξ / δ k / + p ( bv ˆ( av ˆ ξ ( min bv ˆ ˆ av ( ξ + ( ξ + v v ( ( ( ( Th intgral can b put in mor tandard form by changing th intgration variabl to ( ξ ( bv ˆ ( av ˆ ( v c ( ξ ( p + / and chooing th poitiv rang for th variabl ν rult in ( ξ ( p ( p ( p { v } / 3
ξ ( min p dξ ik in / ξ δk ξ / ( ( / p bv ˆ ˆ av δk ξ ( ξ + ( v p ik ξ p ik ξ in δkξ / dξ dξ / δkξ / + ( bv ˆ( av ˆ ξ ( min bv ˆ ˆ av ( ξ + ( ξ + v v ( ( ( ( ik ( ( ( min / p ν ξ ( p ik ( p ν in δ k / p ik dν dν / / δ kp / + ( ν ( ν ik ( p in k ν δ p / ik ( p ν ik dν dν / / δ kp / ( ν ( ξ ( min /( ( ν p Th firt intgral i rlatd to a Hankl function, Gradhtyn and Rhzhic [5] (pag 956 #6, o w can writ ξ ( min p dξ ik in / ξ δk ξ / ( ( / p bvˆ avˆ δk ξ ( ξ + ( v in δ kp / π ik ( ( ( ξ min H ( k ( p H, k ( p δ kp / i p whr th cond trm in th curly brackt rprnt an incomplt Hankl function dfind by tting th lowr limit of th intgration to om poitiv ral valu that i gratr than. Th contribution to th compl probability dnity from th blockd trajctori in thi ca i ρ ( a v ( i/ L( tt p + mc / ( v ( tt v i π NN y A 3/ BS ( p Vk v,;, (/ ( ( ABS p H k p H, k ( p p / i p in δ kp / π ik ( ( ( ξ min ( ( ( p min whr th cofficint A B ha bn introducd for comparion with th ffct of th unblockd trajctori. Whn, on th othr hand, p i gratr than min w will hav / 33
ξ ( min ξ ( p dξ ik in / ξ δk ξ / ( ( / p bv ˆ ˆ av δk ξ ( ξ + ( v dξ in δk ξ / min ik ξ / ( ( / bvˆ ˆ av δk ξ ( ξ + / ( v c th uppr limit of th intgration i rachd bfor th pha trm k -bar ξ bcom tationary with rpct to. Howvr, it pa through zro bfor th uppr limit i rachd, o th in of δk ξ/ dividd by δk ξ/ i tationary at point traddling th origin and vari rathr lowly btwn thm. Thn bcau th pha trm k -bar ξ ocillat incraingly rapidly with a it rcd from th origin, ξ ( min p dξ ik in / ξ δk ξ / ( ( / p bvˆ avˆ δk ξ ( ξ + ( v ξ ( min in δkξ( min / dξ ik ξ / δkξ( min / ( bv ˆ( av ˆ ( ξ + ( v in δkξ( min / dν ik ( ( p δkξ( m / in / ( ξ ( min /( ( ν p in δkξ( min / π ik ( ( ξ min H, k ( p δkξ( min / i p Of cour, ξ( min i l than p and rach a maimum at p a long a th lattr i l than. Th cofficint in thi ca i thrfor ( ( in δk ξ min / π ik ξ min ABS H, k ( p ξ min / i p ( (.6. Intgrating Ovr th Bam on Unblockd Trajctori For unblockd trajctori, auming th quadratic trm in th pha can b nglctd, w hav 34
ρ (, tt ;, v L L W ξ π < < L L < ξ < + W ma d i i( / π ln π a 4 L/ W ξ L/ ξ ( a v ( i/ L( tt p + mc / ( v V NNy 3/ v ( ( L/ + W + ξ( L/ξ ( + ( + min k min(, dk co L + ik k W ink W ξ k Th firt two trm will not contribut unl ξ, which i a function of, tak a valu nar on of th two rang dfining th lit for valu of alo in on of th rang. Sinc i (probably chon to b mall compard to in uch rang, w can u a linar approimation for ξ in thi rang ξ b + ( bv ˆ ( av ˆ ( / v c a ( bv ˆ ( av ˆ ( v ( / ξ ( / + + + Auming i about an ordr of magnitud largr than in th rlvant rang, w that ξ will not vary much during th intgration ovr d, but in ordr to b in on of th allowd rang at all, it mut b that a. v atifi a vˆ ξ ( v bv ˆ for om ξ uch that ξ (+ / and ach fall within th rang dfining on of th two lit. Thu for th trm undr conidration to actually contribut, w that a. v /v mut b ngativ and about a mall compard to a i compard to b. v /v. If that i th ca, w will hav ma min L L L L W ξ W ξ d π < < < < πw a L L ( / L L < ξ < + W < ξ < + W providd ξ (+ / rmain within on of th lit whil vari ovr th rang ( min, ma. Othrwi, th rult will b mallr going to zro if ξ (+ / rmain outid both lit ovr 35
th full rang. Th cond trm contribut two intgral of th form ( a+ ( b+ c ln a+ ( b+ c ( a+ ( b+ c ( ( + ln ( + + ( ( + d ln ( a ( b + c b a b c a b c a b c i 4 ma min L+ W W d ξ + ln ii,,, a 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 ( / 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 / which how that th logarithmic ingulariti of th intgrand will b cancld. Uing bξ /, w can writ Thi till pak nar valu of ξ corrponding to th dg of th lit a can b n in Figur 4, which aum th lit hav qual lin and pac gomtry and (L+W/ qual.. Of cour, th pak would b moothd down if th quadratic trm in th pha had not bn t to zro. Morovr, th contribution from th uppr and lowr lit approimatly cancl cpt nar thir dg. For mall valu of ξ, it i uful to rwrit th third trm in th form ma min ( L W ξ k min(, ma k min(, ik ξ co in co d dk L+ W W W L+ W k k d dk k a k a ik which i appropriat in th limit that k min W/ i mall compard to unity. Actually, inc w ar intrtd in th firt minimum at th targt, k min (L+W/ will b about π/. Thn, vn in th ca that th width of th lit i qual to th pac btwn thm, k min W/ would b only π/4, whr th in i till approimatly linar in it argumnt. Nglcting th mall dpndnc of k min on, on can intrchang th ordr of th intgration to gt min ξ ma min 36
(, ma k min d L+ W ikξ( + / dkw co k a min kmin ma L+ W ikξ d ikξ( / dkco k W a min k min ik ikξ( / ma ξ L+ W W dkco k ( / ikξ / min kmin W L+ W ikξ + / ξ dkco k ( ξ ( in k ( W / ( / k W / Evn if ξ i a larg a, k ξ W/ will b mall in th currnt aumption about k min and W, o th intgration ovr k can b approimatd by ma min (, k min + co kmin W L+ W co d L W dk W k a dk k / ( ξ ( + / ik ξ ( + / ik L+ W L+ W ik + ξ( + / ik + ξ( + / W + ( / L+ W L+ W i + ξ( + / i + ξ + / L+ W L+ W ik min + ξ( + / ik min + ξ( + / W + i( / L W L W + + + ξ( + / + ξ( + / ( kmin Thi form i wll bhavd at mall valu of ξ. Alo, it i almot th am for both lit, inc -bar i alway dividd by. Evn whn ξ i at th cntr of on of th lit ±(L+W/, and k (L+W/π/, w will hav ma min k min(, d L+ W W W W dk W k k i a ± / L+ W ikξ ( + / co min ( At th particular valu of ξ, w find th ral and imaginary part of th third trm partially cancl th firt and cond trm, rpctivly. Whn ξ ±(L+W/ ar both larg compard to W/, it i uful to rwrit th third trm in th form 37
ma min ma min ma min k min(, d dk L+ W W cok ink a k k min(, d dk L+ W W L+ W W in k in k a + k d a ik k min(, 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(, kmin(, kmin(, ξ+ + ξ ξ+ min( ma d du iu du iu du, L+ W W k ξ + iu a min i u i u + i u ξ ik + Thn if ξ fall outid both lit L + W W ξ ± > th intgral combin in pair to giv ma min Figur 4 Imaginary part of th infinit intgral ovr momntum tranfr d dk d a k a i u i u du i u L+ W W L+ W W k min ( (, min(, min, k k ξ+ + ξ + ma L+ W W ik ξ du iu du iu cok ink + min L+ W W L+ W W kmin(, ξ+ kmin(, ξ iu 38
Furthrmor, in th limit L + W W ξ ± th variation in th dnominator bcom ngligibl ovr th rang of th intgration giving ma min min k min(, d dk L+ W W cok ink a k L+ W W k min(, ξ + + ma d iu du + a ξ ik L+ W L+ W i k (, ξ + ( i k, ( ξ ( min(, L+ W W k ξ + L+ W W L+ W W min k min, ξ + min k min, ξ Nglcting alo th mall variation of k min (, on and collcting trm giv ma min k min(, d dk L+ W W cok ink a k ik ma ikmin L/ W ikmin L/ d ξ + ξ ikmin ikmin d + d a L W L W min + i ξ L/ + i ξ L/W + ma ikminξ L/ + W L/ d ikmin L+ W ikmin ikmin ξ d d d a min L W + lit L/ L/W i ξ W hav n that th firt two trm in th compl probability du to an unblockd trajctory ar only ignificant whn ξ i in th vicinity of th lit. Thu, a th magnitud of ξ incra o that ξ ±(L+W/ both bcom larg compard to W/, th contribution from unblockd trajctori bcom ξ du iu 39
ρ ( a v ( i/ L( tt p + mc / ( v (, tt ;, v i(/ π NN y 3/ ( v V + + π ξ i ma ikminξ d i ik in min L + W kmin ( in ξ d iw L W kmin a min L+ W lit kminw π ( / N N y ( a v ( i/ L( tt p + mc / ( v ( v 3/ ma ikminξ d iξ V ( k min a min L+ W ξ ma ik min ξ d W V ( L+ W L+ W in kminw / in k min a min k L+ W π minw / ξ W / / Ecpt for th ponntial trm, th intgrand ar almot contant ovr th mall rang of valu of in both ca. Th avrag valu of ξ in a particular lit i ( bv ˆ ( av ˆ ma ξ d b W + ( min / v c a ma d ( / W ξ + min ma + min ξ+ ξ+ Uing thi notation, th approimat rult can finally b writtn 4
ρ (, tt ;, v i ( / π N N y ( a v ( i/ L( tt p + mc / ( v ( v 3/ ikminξ i Wξ in kminξw / V ( kmin k min W / L W ξ + ξ ikminξ W ( L+ W / in kminξ W / V L+ W in kminw / in k min k min W / kmin / L W ξ π + ξ For comparion of th two trm it i uful to iolat th combination of factor and dfin th complmnt of th Fourir tranform V C C V L+ W in kminw / V ( kmin in kmin π kmin / V ( co kmin ( L/ W co kmin ( L/ + kminπ which ha th am amplitud a th Fourir tranform of th potntial but, bing contructd of coin trm intad of in trm, i ubtantially out of pha with it. Thn introducing A U and B U, a th cofficint of th Fourir tranform and it complmnt in th contribution from th unblockd trajctori, w can writ ρ (, tt ;, v ( a v ( i/ L( tt p + mc / ( v i( / π NNy A 3/ U ξ V k min BU ξ V k min v ( { ( ( ( ( } + Th ratio of th two cofficint B A U U i ( L + W ξ i mall (in abolut magnitud to th tnt that ξ i larg in comparion with th ditanc (L +W/ from th ai of ymmtry to th cntr poition of th lit. By th way, ξ i rlatd to p by 4
( bv ˆ ( av ˆ ξ + v ( ( p and w can that if - p >>3 / thn ξ will b l than -. Thi man that if and - p ar both larg compard to {L+W/, ξ will b alo..7 Probability Dnity of Rcordd Maurmnt Whn ξ i ngativ and larg in magnitud compard to (L+W/, th compl probability contribution from th blockd and unblockd trajctori ar both proportional to th Fourir tranform of th potntial with cofficint for th blockd trajctori and in δ k / A H k ( ( p π ik ( ( B p p p / i A U ikminξ i Wξ in kminξ W / k min W / L+ W ξ ξ for th unblockd trajctori. It will b hown that th contribution of th blockd trajctori dominat whn i chon to b at th firt minimum of th diffraction pattrn, that i k (L +W/ i qual to π/, and alo happn to b larg compard to (L+W/ which in turn i larg compard to W. Conidring th blockd trajctori firt, it i notd that th argumnt of th Hankl function i rathr larg, o w can u th approimation (Gradhtyn and Rhzhic [5] 8.45 #4 on Pag 96 / ik p iπ /4 ( H ( k ( p π k ( p Thrfor, th cofficint for th blockd trajctori bcom ( 4
( p in δ k / π ( p ik p iπ /4 AB p / i π k p in δ k / p / i π k ( p ik p iπ /4 p π / / whr w hav ud k mv k vˆ to dfin k -bar. Nglcting th mall lit dpndnc of th amplitud, th ffct of th pha hift btwn thm bcom / in δ kp / π ikp iπ /4 ik( L+ W p/ ik( L+ W p/ AB ( p ( ( + p / i π k p / ( in δ k / π L+ W δ k k ( p ikpiπ /4 p co δ kp / i π k p Thi nvlop of thi prion dcra lowly a p rtrat from - to largr ngativ valu a long a δk p / rmain mall w δ kp / k p / L+ W W p π k < L+ W In particular, if i chon to b th firt null in th diffraction pattrn, thn thi condition i quivalnt to p L + W < W Thu th rang of (ngativ valu of p in which th um of th cofficint for th blockd trajctori can b ignificant i larg (compard to to th tnt W/(L+W i mall. Within thi rang, th um of th cofficint for th blockd trajctori bcom ( ( L ikpiπ /4 p AB p iδ k cok k p / π + W ( 43
Turning now to th blockd trajctori, thr i alo a pha hift btwn th contribution from th two lit. Actually, both trm in th pha vary and k ξ k ξ ma kξ + W kξ + W W kξ ma min Avraging th mall diffrnc in th magnitud of th two trm, th nt contribution from th two lit du to unblockd trajctori i givn (to lowt ordr in W/ i givn by W L W + ik ξ i Wξ in kξw / W L + W AU cok ξ k L+ W ξw / ξ Th nvlop of thi prion dcra lik (ξ - a ξ rtrat from - toward mor ngativ valu a long a k -bar ξ W/ rmain mall L+ W W ξ kξw / k L + W From it dfinition, w can that ξ ( bv ˆ ( av ˆ ξ + v ( ( p p p i approimatly qual to p whn p i mall compard to. A long a ξ i larg compard to (L+W/, but l than about (+L/W, th nt contribution from th two lit du to th unblockd trajctori will b approimatly 44
i W L+ W W L+ W ik ξ Wξ AU cok ξ L+ W ξ At largr valu of p on th ordr of, ξ bgin to grow quadratically. Thi man that a p rtrat from - toward largr ngativ valu, th contribution from th unblockd trajctori will cut off bfor that of th blockd trajctori do. For ampl, whn - p i largr than 3 /, ξ will b l than -. Th nvlop of th blockd trajctori bgin to dcra lik (ξ - ~ ( op -4 whn ξ / < -(L+W/W which occur whn p L+ W < + W If (L+W/W i larg, th rang in p in which th blockd trajctory contribution rmain ignificant aftr th unblockd trajctori hav cut off / L L + < p < + + W W Evn bfor unblockd trajctori cut off, for ampl whr ξ - th pak valu of thir amplitud / W ξ k( p A δk L+ W π ξ / ξ k ( p k L+ W π ξ p L+ W ξ / L+ W L + W ( ξ / π k AU B ( p / / i mall compard to that of th blockd trajctori. That i, at th firt minimum of th Fourir tranform of th potntial πk -bar(l+w/ i about π. Alo, till auming i tim (L +W/, th ratio of th amplitud will b on th ordr of - tim a function of th rducd variabl ξ / (or p /. For ampl, whn - p i 3 / giving ξ -, th function i about 3 /4 and th blockd trajctori ar till dominant. 45
By hypothi b, th diffrntial probability of rcording th particl at in tim dt whn - p i largr than 3 / i thrfor approimatly ρ (,; tt, v dt / y in δ / p π ( + p NN k L W ( / π V( k δk co 3/ k dt ( v / ( δ k p k p Nt conidr th chang of variabl which thn lad to ρ (,; tt, v dt ( v c ( π ( ( bv ˆ ( av ˆ / p v c ( vˆ v ( tt ( / v v t t dt d p p v NN ( k δ k ( L+ W / ( co v / / y π δ in p / p V k 3/ k dp v ( t t k c δ k p / Th chang of intgration variabl from dt to d p cancl th factor of - p, laving a rlativly impl function at th lowr limit of th intgration. Th intgrand ha a zro at δk p / -π and rmain mall for p / l than about -π(l+w/w. It alo ha zro whr p / i a ngativ odd intgr auming that th firt null in th diffraction pattrn i at. Strictly paking, th factor of t-t i not contant, but olving for it in trm of p ( ( vˆ + ( / ( p / ( ( ( v ( p v t t v c + w that th intgrand will bcom vry mall bfor v (t-t bcom ignificantly diffrnt from. It' apparanc in th dnominator i appropriat for th -dimnional primnt bing conidrd. Whn p approach zro from ngativ valu, k (- p rmain larg, o th compl 46 / /
probability contribution from th blockd trajctori approach a valu that ha th am functional form on p but i dividd by. Howvr, w find ξ ξ p convrg to p. Thi man that th componnt of th compl probability dnity du to th unblockd trajctori that i in pha with th Fourir tranform of th potntial dcra linarly whil th out of pha componnt rmain ubtantially contant. Onc p turn poitiv, though, th contribution from th blockd trajctori will dcra a th lowr limit of th incomplt Hankl function incra from unity toward largr valu whr th intgrand i mallr in magnitud. Th unbound trajctori may thn dominat a p approach, which i th uppr limit of th rang ovr which th pha i tationary at om valu of k, ξ approach /. At thi uppr limit, th contribution from th unblockd trajctori will again b ntially proportional to th Fourir tranform auming / i larg compard to (L+W/. Howvr, th factor of - p rulting from changing th intgration variabl from t to p will thn cancl thi rmaining contribution from th unblockd trajctori. On th othr hand, th rang of arrival tim at which th contribution from th blockd trajctori can b ignificant corrpond roughly to p π δ k p / Sinc δk i roughly W/ mallr than k, thi man th lowr limit of p i on th ordr of -π/k W or p ( L+ W W whn th firt minimum i at. Thrfor - p can b larg compard to if W i mall compard to (L+W/ and till a larg a 5 for W qual to L. Thu it mrg that th blockd trajctori can b ignificant rlativly far from th lit, compard to th obrvation point, and thrfor contribut much mor than th unblockd trajctori. W can now timat th chang in th nrgy of a particl on a trajctory dfind by arriving at th targt poition at tim t givn in trm of p / k ( a v / a v mc c k T T + / + a + a ( / ( / v mca v c v c k a v a k + / a ( v ( v ma 47