Chapter 2. Foundations of quantum mechanics

Transcription

1 I. Basics Caper. Fouaios of uau ecaics Eac observable correspos i uau ecaics o a operaor, wic represes a aeaical operaio. Exaples: ˆ f ˆ f f f Soe ipora operaors i e cooriae represeaio: Cooriae operaor: ˆ Liear oeu operaor: ˆ 4 / π Js i P Hailoia eergy operaor: Hˆ pˆ + V + V 1

2 Te for of a operaor epes o e represeaio. For exaple, e cooriae a oeu operaors i e oeu represeaio becoe: ˆ, pˆ p i p Laer, we will see a rasforaios exis bewee iffere represeaios. Two operaors coue if Ω ˆ, Ωˆ ] Ωˆ Ωˆ Ωˆ Ωˆ [ For exaple, ˆ a pˆ o o coue: f ˆˆ p f f is a arbirary fucio i so ˆ p f ˆ f + f i f f [ ˆ, pˆ ] f f if i i + [ ˆ, pˆ ] i

3 If a fucio φ a saisfies e followig eigeeuaio Ω ˆ φ ωφ, i is calle a eigefucio of e operaor Ωˆ wi e cosa ω as is eigevalue. Exaple: si k k si k So, sik is sai o be a eigefucio of e operaor wi k as e eigevalue. If e eigevalues of wo eigefucios are e sae, ese wo eigesaes are calle egeerae exaple: si k a cos k. I uau ecaics, we oly eal wi Heriia operaors. Ωˆ τ [ Ωˆ τ ] Ωˆ τ Exaple, pˆ is Heriia i i i i [ ] i

4 i. Eigevalues of Heriia operaors are always real. ii. Eigefucios of a Heriia operaor are orogoal: τ if a ca be oralize τ τ 1 Cobiig e wo, e eigefucios are orooral: τ δ were δ if a 1if is e Kroecker ela. iii. Eigefucios of a Heriia operaor for a coplee se, aely, a arbirary wavefucio ca be expresse as a liear cobiaio of e eigefucios: Ψ c Ωˆ ω. Te expasio coefficies, c, wi eeries e cace of e syse o be i a paricular eigesae e superposiio priciple. 4

5 II. Te posulaes of uau ecaics Quau ecaics ca be esablise fro a sall uber of posulaes. Tey cao be erive or prove. Teir valiiy is esablise by e cofiraio of eir preicios. Posulae I: Te sae of a bou uau syse is copleely escribe by a wavefucio Ψ,, were { 1,,... } are e spaial + spi cooriaes of e paricles i e syse. All observables are eerie by Ψ,. Exaple: grou sae wave fucio of a 1D aroic α oscillaor: Ψ Ne Posulae II: Every observable correspos o a Heriia operaor. I e cooriae represeaio, suc a operaor ca be cosruce by expressig ˆ, ˆ p i Exaple: agular oeu operaor: lˆ x yp ˆˆ z zp ˆˆ y i y z z y 5

6 Posulae III Bor s ierpreaio: Te probabiliy for fiig e syse i e volue elee τ a poi is proporioal o Ψ τ. Te probabiliy esiy Ψ is always real a o-egaive, aloug e probabiliy apliue Ψ ca be coplex. Te saisical ierpreaio of e wavefucio reuires a i be sigle value, coiuous, a suare-iegraable: Ψ τ < I fac, wavefucio ca be oralize suc a Ψ τ 1. / a Exaple. Te wavefucio of H ao is Ψ e r were a 5.9 p. Calculae e probabiliy o fi e elecro isie a sall box of 1. p a r a r a. Noralizaio: C Ψ a 4 τ r / a 4π πa e r π π r siθθ φ were we ave use sperical cooriaes: 6

7 x rsiθcosφ, y rsiθsiφ, z r cosθ τ x y z r siθrθ φ a ax e x x a! + 1 Te oralize wavefucio: 1/ 1 r / a e a Ψ, a π Ψ 1 πa e r / a Te probabiliy a r : -6 P Ψ τ p 1. p a a r a : P' p e 1. p Posulae IV Scröiger euaio: Te wavefucio evolves i ie accorig o e euaio: -7 Ψ i Hˆ Ψ 7

8 8 For ie-iepee Hailoia, e ie-epee Scröiger euaio TDSE ca be reuce o a ieiepee for:, Ψ Te TDSE becoes: ˆ H i Divie bo sies by Ψ, we ave ˆ 1 1 H i Sice wo sies epe o iffere variables, ey ave o be cosa E. Hece, e euaio ca be separae io wo: E i, ˆ E H, Te firs as a soluio: / ie e, a e seco oe is calle e ie-iepee Scröiger euaio TISE. Noice a e or of bo Ψ a is e sae because: / / Ψ Ψ ie ie e e Tey are calle saioary saes a / ie e is e pase.

9 Posulae V: If a syse is escribe by e wavefucio Ψ, e ea value of e observable Ω i a series easurees is give by e expecaio value of e correspoig operaor: Ωˆ ˆ Ψ ΩΨτ Ψ Ψτ were e wavefucio ca always be oralize Ψ Ψτ 1. If Ψ is a eigefucio of Ωˆ Ωˆ Ψ ωψ, eac easuree yiels e sae eigevalue ω a Ωˆ ω. If Ψ is o a eigefucio, i ca always be expresse as a liear cobiaio of e eigefucios: so, Ψ Ω ˆ c wi Ωˆ ω c c c c c ω δ ˆ Ω c c ω τ ω τ τ δ Eac easuree always yiels oe of e eigevalues, e cace for a paricular eigevalue ω o appear i a easuree is give by c. 9

10 Suary Quau ecaics Maeaics Measuree Operaio Measurig euipe Operaor Observable Eigevalue Sae of e syse Wavefucio III. Bra a ke oaio I is ofe ore coveie o express wavefucios i bra a ke oaio. bra: ke:,, T raspose: [ ] φ overlap: τ, esiy: iegral wi operaor: Ωˆ Ωˆ τ Eigeeuaio: Ωˆ φ ω φ Orooraliy: δ Expecaio value: Ωˆ ˆ Ω Liear cobiaio: Ψ c 1

11 IV. Heiseberg s uceraiy priciple If wo operaors coue: Ω ˆ, Ωˆ [ 1 ] oe ca always fi siulaeous eigefucios for bo operaors. Tis eas a e correspoig observables ca be siulaeously easure wi arbirary accuracy. Exaple, 1D free paricle V Ψ Ψ E EΨ, or Ψ pˆ a Ĥ coue for a free paricle: pˆ pˆ, So coo eigefucios ca be fou a ey ca be easure siulaeously a accuraely! Soluio of Scröiger euaio: Ψ exp ± ik cosk ± isi k, plae wave were e wave vecor is 1/ k E /. 11

12 Acio of oeu a Hailoia: pˆ Ψ i e ik ik e i ik ke ik kψ Hˆ Ψ e ik ik e ik k Ψ Ψ is eigefucio of bo pˆ a Ĥ, e eigevalues are p k, E k e Broglie s relaio: π π p k λ λ π λ k I is ipossible o specify siulaeously, wi arbirary precisio, bo e oeu a posiio of a paricle, because [ ˆ, pˆ] i. Te cosrai Δp Δ / is Heiseberg s uceraiy priciple, a p a are copleeary observables. Tie a eergy is aoer pair of copleeary observables: ΔE Δ / 1

13 Exaple: Calc. e posiio uceraiy of a 1. kg paricle wi a spee uceraiy of 1-6 /s. Wa if e ass is kg? Δ / Δp / Δv Js/ 1. kg /s However, for a paricle wi e elecroic size Δ 55 Classical - uau correspoece: Classical Quau Sae + p Ψ E. of oio Newo Scröiger Wave aure o yes Ierferece o yes Uceraiy o yes Eergy arbirary ofe uaize Applicabiliy 1, 1 kg 1 p, 1-5 kg 1