130_notes.dvi

28 Time Dependent Perturbation Theory

We have used time independent perturbation theory to find the energy shifts of states and to find
the change in energy eigenstates in the presence of a small perturbation. We will now consider the
case of a perturbation that is time dependent. Such a perturbation can cause transitions between
energy eigenstates. We will calculate the rate of those transitions.

This material is covered inGasiorowicz Chapter 21,inCohen-Tannoudji et al. Chapter
XIII,and briefly in Griffiths Chapter 9.

28.1 General Time Dependent Perturbations

Assume that we solve the unperturbed energy eigenvalue problem exactly:H 0 φn=Enφn. Now we
add a perturbation that depends on time,V(t). Our problem is now inherently time dependent so
we go back to thetime dependent Schr ̈odinger equation.

(H 0 +V(t))ψ(t) =i ̄h

∂ψ(t)

∂t

We willexpandψin terms of the eigenfunctions:ψ(t) =

∑

k

ck(t)φke−iEkt/ ̄hwithck(t)e−iEkt/ ̄h=

〈φk|ψ(t)〉. The time dependent Schr ̈odinger equations is

∑

k

(H 0 +V(t))ck(t)e−iEkt/ ̄hφk = i ̄h

∑

k

∂ck(t)e−iEkt/ ̄h

∂t

φk

∑

k

ck(t)e−iEkt/ ̄h(Ek+V(t))φk =

∑

k

(

i ̄h

∂ck(t)

∂t

+Ekck(t)

)

e−iEkt/ ̄hφk

∑

k

V(t)ck(t)e−iEkt/ ̄hφk = i ̄h

∑

k

∂ck(t)

∂t

e−iEkt/ ̄hφk

Now dot〈φn|into this equation to get the time dependence of one coefficient.

∑

k

Vnk(t)ck(t)e−iEkt/ ̄h = i ̄h

∂cn(t)

∂t

e−iEnt/ ̄h

∂cn(t)

∂t

=

1

i ̄h

∑

k

Vnk(t)ck(t)ei(En−Ek)t/ ̄h

Assume that att= 0, we are in aninitial stateψ(t= 0) =φiand hence all the otherckare equal
to zero:ck=δki.

∂cn(t)

∂t

=

1

i ̄h



Vni(t)eiωnit+

∑

k 6 =i

Vnk(t)ck(t)eiωnkt



