QMGreensite_merged

18.3. SUDDENPERTURBATIONS 299

atamoment(t= 0 −)justbeforetheperturbationisswitchedon. Withthisinitial
condition,wewanttoknowwhatthewavefunctionwillbeatanytimet>0,andin
particulartheprobabilityoffindinganyparticleenergyeigenvalueE′k.
Thisisactuallyquitesimple. Firstofall,atanytimet>0,thesolutiontothe
time-dependentSchrodingerequationhasthegeneralform

ψ(x,t>0)=

∑

n

cnφ′n(x)e−iEnt/ ̄h (18.81)

Thencontinuityofthewavefunctionintime,andparticularlycontinuityattimet=0,
requiresthat
ψin(x,t= 0 −)=

∑

n

cnφ′n(x) (18.82)

Thisequationdeterminesallthecnintheusualway (gotobra-ketnotation,and
multiplyontheleftby〈φ′k|

ck=〈φ′k|ψin(t= 0 −)〉 (18.83)

TheprobabilityofmeasuringenergyEk′ atanytimet> 0 isthen

P(Ek) =

∣∣

∣〈φ′k|ψ(t)〉

∣∣

∣

2

=

∣∣

∣ck

∣∣

∣

2

=

∣∣

∣〈φ′k|ψin(t= 0 −)〉

∣∣

∣

2

(18.84)

18.3.1 Example

Weconsiderthepotential

V(x,t)=

{ 1

2 kx

(^2) t< 0
1
2 k
′x (^2) t≥ 0 (18.85)
SoforanytimettheHamiltonianisthatofaharmonicoscillator(whichisexactly
soluble)butwithspringconstantkatt<0,andk′att>0.
Supposetheparticleisinitsgroundstateattimest<0.Whatistheprobability
thattheparticleisinanexcitedstateatt>0?
Clearlytheprobabilityofbeinginanexcitedstateisisrelatedtotheprobability
P(E 0 ′)ofremaininginthegroundstate,i.e.
Pex= 1 −P(E 0 ′)= 1 −|c 0 |^2 (18.86)
Thegroundstatesbeforeandaftert= 0 are
φ 0 (x) =
(
mk
π^2 ̄h^2
) 1 / 8
exp
[
−

1

2

√

mkx^2 / ̄h

]

φ′ 0 (x) =

(

mk′

π^2 ̄h^2

) 1 / 8

exp

[

−

1

2

√

mk′x^2 / ̄h

]

(18.87)