The Single Particle Path Integral and Its Calculations. Lai Zhong Yuan

Transcription

1 Te Sngle Parcle Pa Inegral and Is Calculaons La Zong Yuan

2 Suary O Conens Inroducon and Movaon Soe Eaples n Calculang Pa Inegrals Te Free Parcle Te Haronc Oscllaor Perurbaon Epansons

3 Inroducon and Movaon

4 In Q.M. a probably aplude assocaed w every eod wereby an even n aure can ake place. Also assocae an aplude w e overall even by addng ogeer e apludes o eac alernave eod. Can be seen va e aous double-sl eperen

5 Te Double Sl Eperen

6 φ Aplude or reacng e deecor wle passng roug ole φ Aplude or reacng e deecor wle passng roug ole Toal aplude o reacng deecor φ φ φ

7 snce Probabl y Probably Aplude Toal Probably o Elecron Reacng Deecor P φ φ Bu any ways o analyze concep o aplude. In s eaple assocae aplude w eac possble oon o elecron ro A o B. Hence oal aplude su o a conrbuon ro eac o e pas

8 A Toug Eperen

9 Eac alernave pa as own aplude Coplee aplude s su o all possble pas. ow suppose: More and ore oles are drlled ll nong ore s le o e screens A eac screen pa o elecron speced by egs D E a wc e elecron passes posons y D y E

10 A Toug Eperen II

11 A Toug Eperen III

12 To eac par o egs corresponds an aplude. eed o ake negral o e apludes over all possble values o. D E Also e e a wc e elecron passes eac pon n space. y Tus a pa s dened roug y Toal aplude su over e aplude or source deecor ravel ollowng all possble pas.

13 Bu ow does one dene a su over a pa? One way s o approae by lne segens le segens

14 Unary o Q.M.: Aplude o e wole pa s produc o e apludes o eac nnesal pa Aplude o propagae ro o n e T s gven n ers o e unary operaor suc a Aplude e HT

15 Dvdng e T no nnesal segens eac o leng T / one as e HT H e Le H Copleeness o eans a d

16 And us Takng a look a one ndvdual acor H H H H HT e e e e d e Π L L H e

17 or ree parcle case along w e copleeness o gves V p / / / ˆ / ˆ p p p p p e e dp p p e dp p p e dp e π π π

18 Te negral over yelds nong a s Gaussan and eployng e relaon ] / [ / ]/ [ / p H e e e π π p a de a π

19 Subsung e negraed epresson no H e epresson or e produc o e yelds e HT π Π d e / [ / ] Takng e connuu l one replaces [ / & and D ] π l Π d T d

20 One as en e pa negral represenaon: e HT D e HT To oban e one sply negraes over all possble pas suc a and T Halonan or a parcle n a poenal V ˆ T d & e HT D e T d[ & V ˆ]

21 Bu and n general As and resorng e Planck s consan e nal epresson s en L V & & dl HT T e D e & S dl T & S HT e D e

22 Soe Eaples n Calculang Pa Inegrals

23 I. Te Free Parcle

24 Te Lagrangan or a ree parcle s noed o be Te ull aplude or kernel s en L & L & & Π d ep l π

25 Te ull epresson can be obaned by subsung e Lagrangan no e acon T dl & negraon. S and perorng e e Alernavely one can also apply e acon prncple along w e Halon euaons or e ree parcle.

26 Takng noe o e deny du π a u ab a e a b u b u y ep b e π a a b y one rs consders e rs negral over n e su o ers n e eponen.

27 Includng wo o e ers one as accordng o e negral deny dsplayed n e prevous slde π ep ep d π π

28 I can be seen a e eec o negraon on e s e replaceens 3 3 bo n e suare roo and n e eponenal 3 negraon : bo n e suare roo and n e eponenal 4 negraon :3

29 Groupng and negrang es one nally as Te nal epresson o e negral s Dsance suared T ep T T π

30 II. Te Haronc Oscllaor

31 Te aronc oscllaor Lagrangan specal case o e uadrac Lagrangan L & b & c e For e aronc oscllaor replacng : / b / a L & ω

32 Te acon o e Lagrangan s o course In general uadrac Lagrangans suc as ar. osc. can be solved by nroducng S τ τ τ y τ classcal Ld raecory were τ.e. were e acon s an ereu S uncanged n e s order τ s oded slgly.

33 Derence beween classcal pa and a possble alernave pa s y. Evdenly y y a b Classcal pa s consan

34 Seng a new negraon varable e Lagrangan can be Taylor-epanded around as y y & & & y L yy L y L y L y L L L & & & & & & & & & & ; ; &

35 Te Taylor epanson ernaes aer e second er snce s a uadrac unconal. Subsung e Lagrangan or e aronc oscllaor yelds or e acon: ; τ L & y y d y L y L d L dl S ; ; ω & & & & & &

36 Perorng negraon by pars and usng S ro e denon o e classcal pa yelds e epresson S S cl d y& ω y Wc gves or e aplude S ; ~ ; ep ; as e pa negral only depends on e

37 Eployng e usual denons o e pa negral one wres To deal w e ulude o negrals one resors o a eod nroduced by Geland and Yaglo a b y y y dy dy ep l ; ~ ω π

38 Geland-Yaglo Meod Te epresson can be wren n ers o ar eleens suc a and us y y y ep ω y y M η ση η ω T y y y

39 Here s e ar dened by and us e aplude can be wren σ c c c O O O O O σ ση η η π T d ep l ~

40 s o e or w real and Heran dagonalzable by a unary ar σ U σ D U σ σ ~ ~ σ σ D were s e dagonal ar o egenvalues o σ. For real egenvecors s also real; one can se ξ Uη. Snce de U s rue a U T η ση ξσ ξ π D d ηe d ξe σ α α π / deσ

41 Te aplude s en wren as σ π σ π π de l de l ; ~ /

42 Te acor Te deernan can be calculaed by consderng runcaed arces ro p c c ' de de de σ σ O O O O O σ '

43 By epandng σ ' n nors can be seen a ey obey e recurson orula p ω p p were and p / ω Rewrng yelds p p p p ωp

44 I one wres ϕ p or a en n e l o ϕ sases d d ϕ ω ϕ w nal values ϕ p dϕ d p p ω as

45 Tus s obaned by solvng e derenal euaon w nal condons a de l ϕ σ ε ω

46 And us one arrves a e aplude or e aronc oscllaor ; ep ; c S π

47 Perurbaon Epansons

48 On general uadrac acons are relavely easy o solve usng e pa negral eod. For general class o poenals perurbaon epanson o pa negral. Sar w a general epresson or e poenal specalze o scaerng proble.

49 a parcle ovng n a poenal as e kernel were s as prevously dened. I e poenal s sall en e poenal par s: V D ep D d V V &! ep V d V d V

50 Subsung no e orgnal epresson: were: V [ ] [ ] [ ] ' ' ' ep ep ep D ds s s V ds s V d dsd s s V d D d b a b a b a & & &

51 Inercangng negraon order and wrng were F s ep F s ds & d V [ s s] D Fs s e su over all pas o e reeparcle aplude; poenal er can be addonally nerpreed.

52 However weged a e s by V s s [ ] Beore and aer s ree parcle. Unary o propagaors su over all pas beween o c & c o can be wren c c

53 Tus or s c F s F c can be wren as F c c V c c d c Subsung no e epresson or c V c c d c d c Can be nerpreed as a ree parcle propagang ro o c s scaered and nally propagaes reely o

54

55 Tus can be wren as V d a c c V d d d d V c c c V d d d d d V c c cv d c c c V τ τ τ τ

56 Tus one can also wre An negral euaon descrbng s known. c V V d c c V c τ V

57 Operang on a waveuncon yelds Subsung e seres epanson o c d d c c V c d b τ ψ V V d b ψ V

58 Te rs er gves e unperurbed waveuncon a e. Callng s er one as wc yelds φ d φ d c c d d d d V d c c V c d c c V c b b τ τ φ τ φ φ ψ

59 Tank You or Your Paence