Double Slits in Space and Time

Transcription

1 Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an on-half cycls long, an so hr ar wo isjoin inrvals of im abou h aks (say) uring which an inoiz lcron can inuc o ravl in on ircion, an on inrval of im abou h rough ha ionizs h lcron in h oosi ircion. A cor is osiion so as o inrc h nrgis of lcrons ioniz by h aks a ims an. A grah of h numbr of lcrons vrsus nrgy hibis inrfrnc frings bcaus hr is uncrainy as o which ak ioniz h aom. In ffc, hr ar slis in im abou an uring which ionizaion occurs, so Paulus s rimn can b inrr as oubl sli rimn wih h usual osiion an momnum rlac by im an nrgy. Mor ails of h rimn ar givn in h Rfrncs scion a h n of h aricl. This aricl aims o giv a simlifi horical mol of Paulu s rimn a a lvl aroria for an ur lvl quanum mchanics cours. To s h sag, a quanum mchanical ramn of h sanar oubl sli in sac rimn is firs rsn. To simlify h mahmaics, unis hav bn chosn such ha h. Doubl Sli in Sac As shown in h figur, aricls iniially in a collima bam ass hrough wo slis, loca a osiions an, an ar c a osiion on a scrn.. y

2 To sar, suos ha only h lf sli, wih osiion, is on, an a aricl is c a locaion on h scrn. Th aricl is fr whil ravling from h sli o h cor, so is vlociy an momnum ar consans of is moion, an v m ( ) v () Th las lin in () follows from h y isanc from h sli o h scrn bing rlaivly larg, i.., >>. Bcaus is consan, quaion () shows ha forcing a aricl hrough h sli an obsrving whr i sriks h scrn is a simulanous masurmn of h aricl s comonn of osiion (h osiion of h sli) an comonn of momnum as i lavs h sli, an as such is subjc o h Hisnbrg uncrainy rlaion. If h sli has infinisimal wih, h sa vcor is hn h osiion ignsa an h numbr of aricls c as a funcion of osiion on h scrn is masur of h momnum robabiliy nsiy ~, whr ~ () ( ) A similar analysis hols for h righ sli. If boh slis ar on, hn ~ ( ) is a masur of numbr of aricls c, bu now h sa vcor is a surosiion of h wo osiion ignsas: ( This rsuls in inrfrnc ffcs. ). (3) If h slis hav fini wih, hn h slis will ach conribu a normaliz coninuous surosiion (ingral) i i i, (4) o h sa vcor, rsuling in. (5) From (5),

3 δ δ, (6) or ohrwis ; or ;. (7) Also from (5), [ ] sin ~ i i i i i i i i i i i (8) Consqunly, [ ] sin cos ~. (9) If slis of wih ar osiion a an, hn h robabiliy isribuions in osiion an momnum sac ar rscivly 5 Doubl Sli in Tim Th abov analysis of h sanar oubl sli rimn was on in rms of coorinas ha ar Fourir ransforms of ach ohr osiion an momnum, wih a oubl sli in osiion an inrfrnc arn for momnum. In som sns, im an nrgy ar Fourir ransforms of ach ohr, which suggss an inrraion of Paulus s rimn in rms of a oubl sli in im an inrfrnc arn nrgy. Saraion of variabls ali o h Schröingr quaion las o wav funcions of h form

4 ie (, ) ( E),. () Usually, h nnc of h saionary sa on h nrgy E is no islay licily, bu his nnc lays an imoran rol in wha follows. To sar h simlifi horical mol of Paulus s rimn, assum ha h aricl cor is loca a. Thn h (, ) of () givs h im nnc of h wav funcion a h cor for an lcron ha has fini nrgy E. In wha follows blow, will always b assum, so h nnc of h wav funcions will b surss. Fr lcrons of various nrgis will b c by h cor, so h wav funcion is a surosiion of h saionary sas of (), an, sinc hr is a coninuum of ossibl nrgis for a fr lcron, h surosiion is an ingral: () ( E) ie E, () whr h facor of has bn inclu for mahmaical convninc. Also for mahmaical convninc, n o ngaiv valus of E by sing ( E) whnvr E. This osn chang h hysics, sinc i givs zro robabiliy for cing a ngaiv nrgy lcron, bu now () bcoms () ( E) ie E, () so ha () is h Fourir ransform of ( E). By h Fourir invrsion horm, ie. (3) ( E) () Th abov analysis of h sanar oubl sli rimn in sac suggss using ; or () (4) ; ohrwis in (), which givs an nrgy robabiliy amliu of ie ie ( E) ( ) sin E (5) E Th nrgy robabiliy nsiy shows an inrfrnc arn an (4) givs a oubl sli in im, E so his simlifi mol sms o rrouc h main rsuls of h rimn. Howvr, (5) os no saisfy ( E) for E, an givs h unwan rsul ha h fr lcron has a 5% chanc of having ngaiv nrgy! Sinc h mahmaical analysis is h sam as for h oubl sli in sac, his is asily sn by rlacing by E in (9), which rsuls in a robabiliy nsiy ha is an vn funcion of E ak abou E.

5 This suggss using h ranslaion E ( E E ) E o chang (5) o ( E) ( E ) E E i ( ( EE ) i EE E ) sin ( E E ), (6) which rsuls in a robabiliy nrgy nsiy ak abou E. Whn E is larg, h robabiliy ha h fr lcron has ngaiv nrgy is ngligibl (bu sill non-zro). Th nrgy wav funcion (5) was riv from h oubl sli in im wav funcion (4), bu (5) was chang o (6) on hysical grouns. Dos (6) also corrson o a oubl sli in im wav funcion? To vrify ha h answr is ys, subsiu (6) ino (): () ie ie () E ( E E ) i( EE ( ) i EE ) sin ( E E ) ie ie i ( E E ) sin E ( E) ie E E ie E (7) Thus, h wo im wav funcions ar h sam u o an ovrall has, an boh giv h sam robabiliy isribuion. (Th rlaion bwn ranslaions an mulilicaion by a has is a gnral rory of Fourir ransforms.) If im slis of wih. ar osiion a. 5 an. 5, hn h robabiliy isribuions in im an nrgy ar rscivly ( E ) In his aml, h robabiliy of h lcron having ngaiv nrgy is.7%, bu arbirarily small robabiliis can b achiv by making E arbirarily larg. Mols wih no ngaiv nrgis ar also ossibl, bu ar a bi mor mahmaically comlica. Rfrncs. h:hysicswb.orgariclsnws93?rss.. h:faculy.hysics.amu.ugg