Wolfram Alpha:

```Time Dependent Perturbation Theory
-----------------------------------

Consider a small time dependent perturbation, ΔH, to the
Hamiltonian, H0.  We assume that the system before the
perturbation is in an eigenstate.  We can write this as:

H'|ψ> = H0|ψ> + ΔH|ψ>   ... 1.

ψ = Σnan(t)un(k)exp(-iωnt)) where un(k) = exp(ikx)

Note that if ΔH is time-independent the sytem will only go into
energy states that have the same energy as the initial state.
These are referred to as stationary states.  If ΔH is a function
of time then the system will go into energy states that differ
by hω from the energy of the initial state.

From the time dependent Schrodinger equation we get:

H'|ψ = ih∂/∂t|Ψ>

Expanding we get:

LHS of 1.

Σnih(∂an(t)/∂t)un(k)exp(-iωnt)) + Σnhωnan(t)un(k)exp(-iωnt))

RHS of 1.

Using H0|ψ> = hωn|ψ>

Σnhωnan(t)un(k)exp(-iωnt)) + ΣnΔHan(t)un(k)exp(-iωnt))

Equating the two gives:

Σnih(∂an(t)/∂t)un(k)exp(-iωnt)) = ΣnΔHan(t)un(k)exp(-iωnt))

If we multiply sides by ψ* = um*(k)(exp(iωmt) we get:

Σnih(∂an(t)/∂t)<um*(k)|un(k)>exp(i(ωmt - ωnt))
= Σn<um(k)|ΔH|un(k)>exp(i(ωmt - ωnt))

Which after applying the orthogonality condition gives:

ih(∂an(t)/∂t) = Σn<um(k)|ΔH|un(k)>exp(i(ωmt - ωnt))

Therefore,

an(t) = (-i/h)<um(k)|ΔH|un(k)>∫dt exp(i(ωmt - ωnt))

= (2/h)<um(k)|ΔH|un(k)>exp(iωt/2) sin(ωt/2)/ω

Where ω = (ωm - ωn)

The transition rate is then:

Γn->m = d|an(t)|2/dt = (2/h2)|<um(k)|ΔH|un(k)>|2sin(ωt)/ω

If there are a continuum of energy states close to ωn then:
+∞
Γi->f = (2/h)|<f|ΔH|i>|2 ∫dω ρ(ω)sin(ωt)/ω
-∞
Noting that E = hω and where ρ(ω) is the density of final states.

Γi->f = (2π/h)|<f|ΔH|i>|2 ρ(ω)

This is referred to as FERMI's GOLDEN RULE.

The transition probability is independent of time (constant).
The <f|ΔH|i> term is called the MATRIX ELEMENT for the transition.
The matrix element can be placed in the form of an integral as:

Mif = ∫ψf*ΔHψidV  V = volume

This of the general form used to find the expectation value of
an operator (observable).

= <ψ|O|ψ> = ∫ψ*Oψ dV

Mif = ∫ψf*ΔHψidV ```