Two-Dimensional Quantum Harmonic Oscillator

Transcription

1 D Qa Haroc Oscllaor Two-Dsoal Qa Haroc Oscllaor 6 Qa Mchacs Prof. Y. F. Ch

2 D Qa Haroc Oscllaor D Qa Haroc Oscllaor ch5 Schrödgr cosrcd h cohr sa of h D H.O. o dscrb a classcal arcl wh a wav ack whos cr h volo follows h corrsodg classcal oo h H.O. las a sgfca rol dosrag h coc of a-classcal corrsodc ca b aalcall solvd boh CM & QM h Schrödgr cohr sa of h D H.O. s a osradg wav ack wh s cr ovg alog h classcal rajcors w wll sar fro h -d. Schrödgr cohr sa for D H.O. o rac h saoar cohr sas ha ar localzd o h corrsodg classcal rajcors 6 Qa Mchacs Prof. Y. F. Ch

3 h Haloa for h soroc D H.O. Carsa coorda: h -d Schrödgr. s: s sarabl: 6 Qa Mchacs Prof. Y. F. Ch Egsas of h D Isoroc Haroc Oscllaor D Qa Haroc Oscllaor H E h Υ Χ E d d d d Υ Χ h h

4 D Qa Haroc Oscllaor Egsas of h D Isoroc Haroc Oscllaor cosl w hav obad dffral. for h D H.O.: h h d d d d Χ E Υ E Χ Υ whr E E E h gfco ad h gval of h D soroc H.O. ar gv / / b % π H H!! E h whr h & h 6 Qa Mchacs Prof. Y. F. Ch

5 D Qa Haroc Oscllaor Egsas of h D Isoroc Haroc Oscllaor h gvals of h D H.O. ar h s of h wo D oscllaor grgs & h gfcos ar h rodc of wo D gfcos I ca b fod ha h gsas!! / / π H H ~ do o rval h characrscs of classcal llcal rajcors v h corrsodc l of larg a br 6 Qa Mchacs Prof. Y. F. Ch

6 D Qa Haroc Oscllaor Egsas of h D Isoroc Haroc Oscllaor 55 Fgr 7. Probabl ds ars of gsas for h D soroc haroc oscllaor 6 Qa Mchacs Prof. Y. F. Ch

7 D Qa Haroc Oscllaor Saoar Cohr Sas of h D Isoroc H.O. I s clar ha h cr of h wav ack follows h oo of a classcal D soroc haroc oscllaor.. cos ; cos Th Schrödgr cohr sa for h D soroc haroc oscllaor s a rodc of wo f srs. Th hod of h raglar aral ss s sd o ak rcs ss o of h rodc of wo f srs. Mahacall h oo of raglar aral ss s calld h Cach rodc of h dobl f srs

8 D Qa Haroc Oscllaor Saoar Cohr Sas of h D Isoroc H.O. Wh h rrsao of h Cach rodc h rs ca b arragd dagoall b grog oghr hos rs for whch has a fd val: /!! Ψ /!! /!!!!

9 fr so algbra Th wav fco abov rrss a of oralzd saoar cohr sa. Saoar Cohr Sas of h D Isoroc H.O. D Qa Haroc Oscllaor C ; Ψ! / C / ~ ;

10 D Qa Haroc Oscllaor Saoar Cohr Sas of h D Isoroc H.O. π /4 π /3 π /.5 π /.5 π /.5 π / Fgr 7. Wav ars of h saoar cohr sas ; for 3 wh dffr vals of h arars ad.

11 D Qa Haroc Oscllaor Saoar Cohr Sas of h D Isoroc H.O. I ca b s ha h cohr sas llc saoar sas. ; corrsod o h Th sroso of wo llc sas wh a has facor h oos sg ca for a sadg wav ar: ; ± ; fgr shows h sadg wav ars corrsodg o h llc sas show fgr abov.

12 D Qa Haroc Oscllaor Saoar Cohr Sas of h D Isoroc H.O. π /4 π /3 π /.5 π /.5 π /.5 π / Fgr 7.3 Sadg wav ars corrsodg o h llc sas show fgr 7..

13 Saoar Cohr Sas of h D Isoroc H.O. D Qa Haroc Oscllaor afsl rvals h rlaosh bw h Schrödgr cohr sa ad h saoar cohr sa. rrss h robabl of fdg h ss h llc saoar sa wh ordr. Th robabl dsrbo s dcal o h Posso dsrbo wh h a val of C ; Ψ C! C > <

14 glar Mo D Cofd Sss D Qa Haroc Oscllaor aglar o of a classcal arcl s a vcor a glar o s h ror of a ss ha dscrbs h dc of a objc sg abo h o r o ra sg classcall. For h oo of a classcal D soroc haroc oscllaor h aglar o abo h z-as ca b fod o b dd of : r L cos cos h h s s d d d d h h s h

15 glar Mo D Cofd Sss D Qa Haroc Oscllaor I a chacs h aglar o s assocad wh h oraor ha s dfd as For D oo h aglar o oraor abo h z-as s Th cao val of h aglar o for h saoar cohr sa ad -dd wav ack sa whch ar show blow : L r L z L / ~ ; C ; Ψ

16 D Qa Haroc Oscllaor glar Mo D Cofd Sss Th oso ad o oraors for h haroc oscllaor ca b rs of h crao ad ahlao oraors. L?? z h a a a a a a a a????????? h a a a a

17 glar Mo D Cofd Sss D Qa Haroc Oscllaor Th rors of h crao ad ahlao oraors : a a / / ~ ; / / ~ ; a a

18 glar Mo D Cofd Sss D Qa Haroc Oscllaor Wh h orhooral ror of h gsas : a a / / ; ; a a / / ; ;

19 D Qa Haroc Oscllaor glar Mo D Cofd Sss Usg h ror W ca oba ad ; L ; z??? ; h a a a a ; h s h

20 Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs D Qa Haroc Oscllaor Th -dd Schrödgr ao for a D haroc oscllaor wh cosra frcs ca grall gv b s h coo facor of h frcs b ad ad ad ar rlav r grs Th gfco ad h gval of h D haroc oscllaor wh cosra frcs ar gv b E h!! ~ / / H H π E h h h h

21 Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs D Qa Haroc Oscllaor Th gfco s sarabl so h corrsodg Schrödgr cohr sa ca b rssd as h rodc of wo D cohr sas: Ψ / / / / / / / / / ~!!!!!! H H π π

22 D Qa Haroc Oscllaor Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs I s clar ha h cr of h wav ack follows h oo of a classcal D soroc haroc oscllaor.. cos ; cos Th s of sas wh dcs las ag ca b dvdd o sbss characrzd b a ar of dcs gv b od ad od Schrödgr cohr sa ca b rwr as Ψ!! / % [ / /]

23 Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs D Qa Haroc Oscllaor Th D Schrödgr cohr sa s dvdd o a rodc of wo f srs ad wo f srs Wh h rrsao of h Cach rodc h rs ca b arragd dagoall b grog oghr hos rs for whch : ~ Ψ ~ ]! [! ] [ / ~ ]! [! ] / / [ / ] / / [ /

24 Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs D Qa Haroc Oscllaor Ths saoar cohr sas ar hscall cd o b assocad wh h Lssajos rajcors. Th or dcs ad ssall do o affc h characrscs of h saoar sas. Icldg h oralzao codo h saoar cohr sas Carsa coordas ar gv b ~ ]! [! ] [ ]! [! ; /

25 D Qa Haroc Oscllaor Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs Th saoar cohr sas assocad wh h Lssajos rajcors ar h sroso of dgra gsas wh h rlav ald facor ad has facor. Th rlav ald facor ad has facor h saoar cohr sas ; ar lcl rlad o h classcal varabls h grgs of h saoar cohr sas ar fod o b E [ / / ] h ;

26 D Qa Haroc Oscllaor Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs hr fgrs dc h coarso bw h a wav ars ; ad h corrsodg classcal rodc : orbs for o b : 3: ad 4 : 3 rscvl. Thr dffr has facors. 3π ad. 6π ar dslad ach fgr. Th bhavor of h a wav ars all cass ca b fod o b rcs agr wh h classcal Lssajos fgrs.

27 D Qa Haroc Oscllaor Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs a b c a b c

28 D Qa Haroc Oscllaor Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs a b c a b c

29 D Qa Haroc Oscllaor Qa Saoar Cohr Sas for Classcal Lssajos Prodc Orbs a b c a b c