First-order time-dependent perturbation theory

Derivation

(a)  Expanding the state vector|ψ(t)>

Let H(t) = H0 + W(t). 
Let {|Φn>} be an orthonormal eigenbasis of H0,  H0n> = Enn>.
Let  ωmn = (Em - En)/ħ and Wmn(t) = <Φm|W(t)|Φm>.
Assume that at t = 0 the system is in the state |Φi>.

We can expand |ψ(t)> in terms of the eigenfunctions od H0,
|ψ(t)> = ∑ncn(t)exp(-iEnt/ħ)|Φn>.

The Schroedinger equation yields
iħ∂/∂t(∑ncn(t)exp(-iEnt/ħ)|Φn>) = (H0 + W(t))∑ncn(t)exp(-iEnt/ħ)|Φn>
or
iħ(∑n(dcn(t)/dt)exp(-iEnt/ħ)|Φn>) = W(t)∑ncn(t)exp(-iEnt/ħ)|Φn>.

Taking the inner product with <Φm|exp(iEmt/ħ) yields
dcm(t)/dt = (-i/ħ)∑nm|W(t)|Φn>cn(t)exp(iωmnt).

Expressing this equation in integral form we have
cm(t) = cm(0) + (-i/ħ)∑n0tm|W(t')|Φn>cn(t')exp(iωmnt')dt'.

At t = 0 we have cn(0) = δni
Assume that at t > 0 but very small we have ci ≈ 1 and cn≠i very small.  Then
cm≠i(t) ≈ (-i/ħ)∫0tm|W(t')|Φi>exp(iωmit')dt'.

The probability of finding the system in the state |Φf> (f ≠ i) at time t is
Pif(t) = |cf≠i(t)|2= (1/ħ2)|∫0texp(iωfit')Wfi(t')dt'|2.


(b)  Changing the representation to the interaction picture

In the Schroedinger picture let H(t) = H0 + W(t)
and let U0(t,t0) = exp(iH0t/ħ) be the evolution operator of a system with Hamiltonian H0.
Let {|Φn>} be an orthonormal eigenbasis of H0,  H0n> = Enn>.
Let  ωmn = (Em - En)/ħ and Wmn(t) = <Φm|W(t)|Φm>.
Assume that at t = 0 the system is in the state |Φi>.

Define the state vector |ΨI(t)> in the interaction picture from the state vector in the Schroedinger picture through
I(t)> = U0(t,t0)|ΨS(t)> =  exp(iH0t/ħ)|ΨS(t)>.
The state vector in the interaction picture evolves according to
iħ∂|ΨI(t)>/∂t = WI(t))|ΨI(t)>,
with WI(t) = exp(iH0t/ħ) WS(t) exp(-iH0t/ħ).

We can rewrite this differential equation in the form of an integral equation.
d|ΨI(t)> = (iħ)-1WI(t))|ΨI(t)>dt.
t0t d|ΨI(t)>  = (iħ)-1t0t WI(t')|ΨI(t')>dt'.
I(t)> = |ΨI(t0)> + (iħ)-1t0t WI(t')|ΨI(t')>dt'
This integral equation can be solved by iteration.
The ket |ΨI(t)> can be expanded in a power series of the form
I(t)> = {I + (iħ)-1t0t dt'WI(t') + (iħ)-2t0t dt'WI(t') ∫t0t' dt''WI(t'') + ... }|Ψ0(t)>.

Assume W(t) is a small correction to H0, and W(t) = 0 for t < 0.  Then |ΨI(0)> = |ΨS(0)>.  Neglecting higher order terms we have
I(t)> = |ΨI(t0)> + (iħ)-1t0t WI(t')|ΨI(0>dt'.
The probability Pif(t) of finding the system in the eigenstate |Φf> (f ≠ i) of H0 at time t is |<ΦfI(t)>|2.  (The predictions of quantum mechanics are independent of the representation.)
fI(t)> = <Φfi> + (iħ)-10t f|WI(t')|Φi>dt'
= (iħ)-10t exp(i(Ef - Ef)t/ħ) <Φf|WS(t')|Φi>dt'.

The probability of finding the system in the state |Φf> (f ≠ i) at time t is
Pif(t) = (1/ħ2)|∫0texp(iωfit')Wfi(t')dt'|2.