First-order time-dependent perturbation theory

Derivation

(a)  Expanding the state vector|ψ(t)>

Let H(t) = H₀ + W(t). 
Let {|Φ_n>} be an orthonormal eigenbasis of H₀,  H₀|Φ_n> = E_n|Φ_n>.
Let  ω_mn = (E_m - E_n)/ħ and W_mn(t) = <Φ_m|W(t)|Φ_m>.
Assume that at t = 0 the system is in the state |Φ_i>.

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

The Schroedinger equation yields
iħ∂/∂t(∑_nc_n(t)exp(-iE_nt/ħ)|Φ_n>) = (H₀ + W(t))∑_nc_n(t)exp(-iE_nt/ħ)|Φ_n>
or
iħ(∑_n(dc_n(t)/dt)exp(-iE_nt/ħ)|Φ_n>) = W(t)∑_nc_n(t)exp(-iE_nt/ħ)|Φ_n>.

Taking the inner product with <Φ_m|exp(iE_mt/ħ) yields
dc_m(t)/dt = (-i/ħ)∑_n<Φ_m|W(t)|Φ_n>c_n(t)exp(iω_mnt).

Expressing this equation in integral form we have
c_m(t) = c_m(0) + (-i/ħ)∑_n∫₀^t<Φ_m|W(t')|Φ_n>c_n(t')exp(iω_mnt')dt'.

At t = 0 we have c_n(0) = δ_ni. 
Assume that at t > 0 but very small we have c_i ≈ 1 and c_n≠i very small.  Then
c_m≠i(t) ≈ (-i/ħ)∫₀^t<Φ_m|W(t')|Φ_i>exp(iω_mit')dt'.

The probability of finding the system in the state |Φ_f> (f ≠ i) at time t is
P_if(t) = |c_f≠i(t)|²= (1/ħ²)|∫₀^texp(iω_fit')W_fi(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 {|Φ_n>} be an orthonormal eigenbasis of H₀,  H₀|Φ_n> = E_n|Φ_n>.
Let  ω_mn = (E_m - E_n)/ħ and W_mn(t) = <Φ_m|W(t)|Φ_m>.
Assume that at t = 0 the system is in the state |Φ_i>.

Define the state vector |Ψ_I(t)> in the interaction picture from the state vector in the Schroedinger picture through
|Ψ_I(t)> = U₀^†(t,t₀)|Ψ_S(t)> =  exp(iH₀t/ħ)|Ψ_S(t)>.
The state vector in the interaction picture evolves according to
iħ∂|Ψ_I(t)>/∂t = W_I(t))|Ψ_I(t)>,
with W_I(t) = exp(iH₀t/ħ) W_S(t) exp(-iH₀t/ħ).

We can rewrite this differential equation in the form of an integral equation.
d|Ψ_I(t)> = (iħ)^-1W_I(t))|Ψ_I(t)>dt.
∫_t0^t d|Ψ_I(t)>  = (iħ)^-1∫_t0^t W_I(t')|Ψ_I(t')>dt'.
|Ψ_I(t)> = |Ψ_I(t₀)> + (iħ)^-1∫_t0^t W_I(t')|Ψ_I(t')>dt'
This integral equation can be solved by iteration.
The ket |Ψ_I(t)> can be expanded in a power series of the form
|Ψ_I(t)> = {I + (iħ)^-1∫_t0^t dt'W_I(t') + (iħ)^-2∫_t0^t dt'W_I(t') ∫_t0^t' dt''W_I(t'') + ... }|Ψ₀(t)>.

Assume W(t) is a small correction to H₀, and W(t) = 0 for t < 0.  Then |Ψ_I(0)> = |Ψ_S(0)>.  Neglecting higher order terms we have
|Ψ_I(t)> = |Ψ_I(t₀)> + (iħ)^-1∫_t0^t W_I(t')|Ψ_I(0>dt'.
The probability P_if(t) of finding the system in the eigenstate |Φ_f> (f ≠ i) of H₀ at time t is |<Φ_f|Ψ_I(t)>|².  (The predictions of quantum mechanics are independent of the representation.)
<Φ_f|Ψ_I(t)> = <Φ_f|Φ_i> + (iħ)^-1∫₀^t <Φ_f|W_I(t')|Φ_i>dt'
= (iħ)^-1∫₀^t exp(i(E_f - E_f)t/ħ) <Φ_f|W_S(t')|Φ_i>dt'.

The probability of finding the system in the state |Φ_f> (f ≠ i) at time t is
P_if(t) = (1/ħ²)|∫₀^texp(iω_fit')W_fi(t')dt'|².