300 CHAPTER18. TIME-DEPENDENTPERTURBATIONTHEORY
Then
c 0 = 〈φ′ 0 |φ 0 〉=(
m
π^2 ̄h^2) 1 / 4
(kk′)^1 /^8∫
dxexp[
−1
2
(
√
mk+√
mk′)x^2 / ̄h]=
√
2
(kk′)^1 /^8
(√
k+√
k′)^1 /^2(18.88)
sothat
P(E′ 0 )= 2(kk′)^1 /^4
√
k+√
k′(18.89)
Itsinterestedtopluginsomenumbers.Supposethatthespringconstantchanges
drasticallyatt=0,byafactorof16,i.e.
k′=k
16(18.90)
Wemightexpectthatsuchalargechangewouldbeverylikelytokicktheparticle
outofthegroundstate. butinfact,onefindsinthiscasethatP(E′ 0 )=^45 ,sothat
thereisactuallyonlya20%chancethattheparticleisfoundinanexcitedstateafter
timet=0.