18.2. ADIABATICPERTURBATIONS 295
SubstitutingthisexpressionforRand∆Rinto(18.54)
n =
1
2
π
(
2 mL^2
π^2 ̄h^2
)
E
1
2
(
2 mL^2
π^2 ̄h^2
) 1 / 2
E−^1 /^2 ∆E
=
π
4
(
2 mL^2
π^2 ̄h^2
) 3 / 2
E^1 /^2 ∆E (18.57)
andfromthiswereadoffthedensityoffinalstates
g(E)=
π
4
(
2 mL^2
π^2 ̄h^2
) 3 / 2
E^1 /^2 (18.58)
18.2 Adiabatic Perturbations
Nextweconsider thesituationwheretheperturbationisturnedon veryslowly,so
thatdV′/dtisnegligible(inasensetobeexplainedshortly). Then,startingfromthe
firstorderterm
c(1)k (t)=
1
ih ̄
∫t
−∞
dt′Vkl(t′)eiωklt
′
(18.59)
weintegratebyparts
c(1)k (t) =
1
i ̄h
∫t
−∞
dt′Vkl(t′)
1
iωkl
∂
∂t′
eiωklt
′
= −
1
̄hωkl
∫t
−∞
dt′Vkl(t′)
∂
∂t′
eiωklt
′
= −
1
̄hωkl
[
Vkl(t)eiωklt−
∫t
−∞
dt′
∂Vkl
∂t′
eiωklt
′
]
(18.60)
Thenassumingthetimederivativeofthepotentialisnegligible,
c(1)k (t) ≈ −
1
̄hωkl
Vkl(t)eiωklt
= −
〈φk|V(t)|φl〉
Ek(0)−El^0
eiωklt (18.61)
Thesolutiontothetime-dependentSchrodingerequation,tofirstorder,is
ψ(x,t)=ψl(x,t)+λ
∑
k(=l
〈φk|V(t)|φl〉
El(0)−E^0 k
eiωkltψk(x,t) (18.62)
andusing
ψk(x,t)=φk(x)e−iωkt (18.63)