QMGreensite_merged

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)