(a) Expanding the state vector|ψ(t)>

Let H(t) = H_{0 }+ W(t).

Let {|Φ_{n}>}
be an orthonormal eigenbasis of H_{0}, H_{0}|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 H_{0},

|ψ(t)> = ∑_{n}c_{n}(t)exp(-iE_{n}t/ħ)|Φ_{n}>.

The Schroedinger equation yields

iħ∂/∂t(∑_{n}c_{n}(t)exp(-iE_{n}t/ħ)|Φ_{n}>)
= (H_{0 }+ W(t))∑_{n}c_{n}(t)exp(-iE_{n}t/ħ)|Φ_{n}>

or

iħ(∑_{n}(dc_{n}(t)/dt)exp(-iE_{n}t/ħ)|Φ_{n}>)
= W(t)∑_{n}c_{n}(t)exp(-iE_{n}t/ħ)|Φ_{n}>.

Taking the inner product with <Φ_{m}|exp(iE_{m}t/ħ)
yields

dc_{m}(t)/dt = (-i/ħ)∑_{n}<Φ_{m}|W(t)|Φ_{n}>c_{n}(t)exp(iω_{mn}t).

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_{mn}t')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_{mi}t')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/ħ^{2})|∫_{0}^{t}exp(iω_{fi}t')W_{fi}(t')dt'|^{2}.

(b) Changing the representation to the interaction picture

In the Schroedinger picture let
H(t) = H_{0} + W(t)

and let U_{0}(t,t_{0}) = exp(iH_{0}t/ħ) be the
evolution operator of a system with Hamiltonian H_{0}.

Let {|Φ_{n}>}
be an orthonormal eigenbasis of H_{0}, H_{0}|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)>
= U_{0}^{†}(t,t_{0})|Ψ_{S}(t)> =
exp(iH_{0}t/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(iH_{0}t/ħ) W_{S}(t) exp(-iH_{0}t/ħ).

We can rewrite this differential equation in the form of an integral equation.

d|Ψ_{I}(t)> = (iħ)^{-1}W_{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}(t_{0})> + (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'')
+ ... }|Ψ_{0}(t)>.

Assume W(t) is a small correction to H_{0}, 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}(t_{0})> + (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 H_{0} 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/ħ^{2})|∫_{0}^{t}exp(iω_{fi}t')W_{fi}(t')dt'|^{2}.