THE TUNNELING EFFECT AND SOME IMPLICATIONS. J.R. Croca

Transcription

1 HE UNNELNG EFFEC ND SOME MPLCONS J.R. Coca Dpaamno d Físca Facudad d Cêncas Unvsdad d Lsboa Campo Gand Ed. C Lsboa Pouga ma: coca@fc.u.p bsac: s shown ha h unnng ffc can mahmacay b dscbd by h unnng opao. Snc h unnng opao s nvb s possb o spcfy h fom of h wav nsd h ba and fom av a h ncdn wav n h nonunnng gon. n asy amp of hs s psnd. On h oh hand hs opao aows us o bud a no-movng ocad sucu nsd a unnng ba a any dsanc as fa as on wsh ndpndn of m. hs sms o mpy an nsananous ans m nsd h ba o n oh wods an nfn vocy. h ova pocss s h appd o a phoonc ocad sucu n h so-cad cassca fusad oa fcon wh a spac bsds h cysa can b an as h unnng ba. h pocss basd on wav anayss avods h pobms asd by Fou nonoca and nonmpoa paadgm. ovcoms h dffcu fo no sayng mpossb pobm of h dfnon of h u vocy of a wav oh han h on of h hamonc pan wav. Ky wods: fundamna quanum physcs unnng unnng opao supumna vocs opcs wav oca anayss.. NRODUCON h unnng ffc has bn sudd fo many yas bu un now no on as fa as now has v d o fnd an opao o dscb mahmacay. h advanag of an opao fo dscbng hs ffc sus fom h fac ha aows us o pass fom h nowdg of h fom of h wav n on gon o psson of h wav n h oh gon. mos smp amp of s shown. By spuang h anayc fom of h wav n h unnng gon w dcy av a h wav n h noma gon. Fuhmo h unnng opao aows us o sudy n a vy asy and nuv way h comp pobm of h supumna ans ms. Rcny many pmns [] hav bn don showng whou any magn of doubs ha h puss whch coss h unnng ba av bfo h ons gong hough h a. Evybody ags wh h sus of h pmns. h quson s f hy mpy a Ppn

2 supumna vocy o no? Snc a cacuaons a don und Fou nonoca and nonmpoa paadgm [] s vy dffcu fo no sayng mpossb o dfn h vocy of a wav. Ony n h cas of an nfn boh n m and n spac hamonc pan wav h pobm offs no dffcuy. n hs nonoca paadgm ony h nfn hamonc pan wav has a u vocy and fquncy. oh fn wavs a no mo han combnaons of hs nfn wavs. hfo any of hs fn wavs has n pncp as many vocs and fquncs as h numb of hamonc pan wavs ha ma. Snc hs hamonc pan wavs a spad ov h who spac and m hs had o n hs paadgm whch s h vocy of a fn wav [3]. Snc h componns ha ma h pus f a spac and m paccay anyhng s possb. h wds hngs a hn possb fo nsanc oacons n m. ha s a pus may av a a dsnaon po bng poducd by h souc! Suaons of hs nd a no nw n physcs. W hav jus o ca h dscpon and pdcon n gophyscs of ahquas. h us of Fou anayss mps n can cass ha h ssmc puss av bfo h acua ahqua oo pac. Oh ms h pcusos sad bfo h fomaon of h Eah sf! Snc gophyscss a pop facng vyday pacca conc pobms hy nd o hav h f pacd on sod gound so hy dscad as a nusanc hs aban suaons sung fom h us of Fou goba paadgm. was pcsy o avod hs gua suaons ha h gophyscs Jan Mo cad n h ghs h concp of fn wav fom whch wav oca anayss spou [4]. H w foow a mo gna appoach[] nspd n h oca anayss fo h unnng pocss. hs mhod aows us o ovcom h abov concpua and pacca dffcus asd by h usua nonoca and nonmpoa Fou paadgm whch n hs cas s ovcom by a oca paadgm.. HE UNNELN OPEROR h unnng ffc mans h passag fom a noma gon of spac o oh chaacd by a pona wh ngy ga han h ngy of h ncdn pac. hs acon aso can mahmacay b dscbd by h foowng opaon f N unnng f f wh sandng fo h unnng opao and f bng a gnc souon of h wav quaon. hs mans ha h passag fom a noma gon o a unnng on s o b dscbd by changng n h souon of h wav quaon h cassca vocy by an magnay vocy o quvany: by. h passag fom a unnng gon o a noma on s don und h sam unnng opao f unnng f f. N N Ppn

3 w b shown ha n can condons h m dos no appa n h mahmaca psson dscbng h no-movng ocad sucu. hs mans ha h no-movng sucu can b ocad a any pon nsd h ba ndpndny of m. ccpng hs concuson by s sagh vau fogng ha h mahmaca fomuas w us a ony wough b o wos appomaons fo dscbng Ray woud mpy an nsananous m ansmsson. 3. UNNELNG WVES WH CONSN NENSY Bfo showng wh a conc amp ha h unnng opao has ndd h abov pops w sha appy o a mos smp amp. h su s as w sha s n a can sns h on o b pcd. By appyng h unnng opao o h wav souon o h cassca wav quaon of h fom w oban a wav whch nnsy nsd h unnng gon s consan a aong h as * C C * C 3 h ncdn wav n h noma gon N f 4 s a cscn wav fom m. o 0 wh a mamum a pon o and dcasng n nnsy - 0 Fg. Po of h ncdn wav Ppn 3

4 Fo h fcd wav h cas s symmca snc h funcon sas dcasng fom h o ogn o mnus nfny 0 o. Summng up vyhng happns as f h bounday bwn h wo gons bhavd a mo gvng ogn o a vua fcd wav as ndcad n Fg.. nnsy - 0 Fg. ncdn and vua fcd wav Snc w a dang n pacc wh a dcasng funcon s dvav mus b ngav. n hs condon ang n accoun h abov consdaons h oa wav n h noma gon s o b wn N B 5 and n h unnng on C. 6 By mposng h condon of connuy of h funcon n a doman w hav a h bounday 0 gvng 0 0 and N N B B C C o B C B C 8 ha s B 9 wh B and Ppn 4

5 C. 0 h po of h nnss of h ncdn and unn wav nnsy a a vy sho m na o s shown n Fg.3 nnsy Noma Rgon unnng Rgon - 0 Fg.3 Po of h ncdn and unn wav nnsy a a vy sho m na o hs su as w hav sad ough o b pcd. Snc a wav of consan ampud a pan wav ncdn on an nfn unnng ba gvs s nsd o an vanscn wav h nvs opaon s aso u. n vanscn wav n h noma gon gvs ogn o a wav of consan nnsy 3 n h unnng gon. 4. HE UNNELNG OPEROR N HE CSE OF CLSSCL OL FRUSRED REFLECON hs cacuaons can b found n any good boo of opcs [5] nvhss fo h sa of cay som sps w b shown. Consd Fg.4 showng wo opca mdums wh facv nds n and n such ha n >n and a bam of gh wh an ncdn ang. Fuhmo fo smpcy asons w assum ha h scond mdum s h a hfo w hav n. n n 0 n h gon h wav vco can b wn Fg.4 Rfacon of gh Ppn 5

6 . cos sn h pobm now s o pss hs funcon n ms of h ang of ncdnc. Fom Sn facon aw and ang n consdaon ha w a nsd n h unnng suaon s ncssay o mpos h oa fcon condon. hfo af som va cacuaons w av a. sn sn hs psson ndcas ha h componn of h vco dcon has h gnc fom aong h pnaon wh sn sn. 3 hs cay shows ha h passag fom a noma gon o oh und unnng condons s ndd dscbd by h unnng opao such ha unnng. 4 h sam concuson coud aso hav bn b achd f nsad of a cassca cacuaon w had don a quanum on fo h cas of pnaon of a pona ba [6] whn h ngy of h ncdn pac s ss han h ngy of h ba. 5. NO-MOVNG LOCLZED SRUCURE NSDE UNNELNG BRRER Pobaby h mus happn o b many pocsss fo obanng a no-movng ocad sucu nsd a unnng ba. Nvhss w sha f h ony h on ha uss a wav wh ampud of h yp. 5 n h cas und consdaon s Fg.4 w hav fo h wav vco cos sn sn sn 6 o mmbng ha Ppn 6

7 sn cos 7 and Sn aw h wav vco can b wn sn sn 8 und such condons h wav souon of h wav quaon n h wo gons has h fom sn sn C B 9 hs pssons ndca ha h componns a qua. hs su shoud b pcd snc aong ha dcon h s no dsconnuy. h pssons 9 can aso b wn 9 sn sn n od o f h vaus of h consans s ncssay o mpos h cusomay connuy bounday condons hs cacuaons adng o C B C B whch gv B and C 3 Ppn 7

8 wh. 4 B n od o noduc a ocad no-movng sucu nsd h ba s ncssay o fnd h gh fom fo h paam. n such condons s convnn o hav fo h ampud of h ncdn wav 5 hs shows ha w nd o hav. 6 h unnng opao appd o 5 gvs 7 wh. 8 Dvopng 7 w go. 9 h vau of mus b such as o psv h ocad sucu. 30 Fom 9 can b sn ha fo h ampud of h fcd wav w hav 3 wh. 3 Wng h h quas 6 3 and 8 Ppn 8

9 whch shows ha 33 and. 34 s no convnn o ma bcaus woud mpy 0 hus conducng o h oss of h ocad sucu. possb choc sus fom mang 0 35 adng o 36 gvng 37 Fo hs cas h funcon 9 n h unnng gon assums h fom. 38 n h cas of h noma gon w hav fom 8 h ncdn ampud 39 and fo h fcd. 40 Fnay oong a 9 3 and 38 w a aowd o w fo h wav n h unnng gon Ppn 9

10 sn 4 gvng fo h nnsy Rscang such ha 43 fomua 4 bcoms whch mamum 4 M 44 aong h as s cnd a h pon. hs facs cay show ha sang fom an ncdn wav of h fom s 9 and 43 sn 45 o sn 45 s possb o oban 4 a gaussan sucu ocad a a dsanc fom h ogn. 0 4 Snc hs paam s as ag as on wsh hs mans ha s acuay possb o pac a ocad sucu n paccay no m a any pon w. ncdnay bcaus Ppn 0

11 h poson of h ocad sucu dos no dpnd on m h hoca da ansmsson s n hs appoach nsananous. n od o ma h po of h nnsy dsbuon n h wo gons s convnn o cacua h oa nnsy n h noma gon. Fom h fom of h ncdn and fcd wavs s 9 and w can w sn * sn 44 whch af som cacuaons gv 0 sn cos 4 ch 45 n appoma po of h nnsy aong h as a a m 0 s shown n Fg.5. 0 { nnsy oma N n Rgo unnng Rgon Fg.5 ppoma psnaon of h nnsy dsbuon n h wo gons fo h m 0. 6 Concuson hs hoca sus sm o ndca ha n unnng condons h usua avsc vocy m c υ bas down aowng paccay nsananous moon. h pobm now s o fnd ou f hs sus a mo han m hoca concusons and hav ndd any a physca cospondnc. Ppn

12 CKNOWLEDGEMENS hs wo was don und h sach pojc POCSOC of h Cno d Fosofa das Cêncas da Unvsdad d Lsboa CFCUL wh suppo and appova of FC and POC bng paay fnancd by communaan fund FEDER. REFERENCES S fo nsanc: G. Nm. Ends and H. Sp n Wav and Pac n Lgh and Ma dd by Van d Mw and. Gaucco Pnum Nw Yo 993; G. Nm Phys. Rv B ;. Ends Rcam Found. Phys JR Coca owads a nonna quanum physcs Wod Scnfc London E. H. Haug and J.. Søvnng Rv. Mod. Phys ; V. S. Ohovsy and E. Rcam Phys. Rp Gossmann and J. Mo SM J. Mah. na ; C.K Chu n noducon o Wavs cadmc Pss N.Y M. Bon and E. Wof Pncps of Opcs Pgamon Pss Nw Yo 983; E. Hch and. Zajac Opcs ddson-wsy Radng Mass D. Bohm Quanum hoy Pnc-Ha Nw Yo 955. Ppn