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.