Define the unperturbed Hamiltonian by H 0 , the time dependent perturbing Hamiltonian by H 1 and total Hamiltonian by H .
The eigenstates of the unperturbed Hamiltonian are assumed to be
H = H 0 + H 1 H 0 | k ⟩ = E ( k ) | k ⟩ In the interaction picture, the state ket is defined by
| k ( t ) ⟩ I = e i H 0 t / ℏ | k ( t ) ⟩ S = ∑ k ′ c k ′ ( t ) | k ′ ⟩ By a Schrödinger equation, we see
i ℏ ∂ ∂ t | k ( t ) ⟩ I = H 1 I | k ( t ) ⟩ I which is a Schrödinger-like equation with the total H replaced by H 1 I .
Solving the differential equation, we can find the coefficient of n-state.
c k ′ ( t ) = δ k , k ′ − i ℏ ∫ 0 t d t ′ ⟨ k ′ | H 1 ( t ′ ) | k ⟩ e − i ( E k − E k ′ ) t ′ / ℏ where, the zeroth-order term and first-order term are
c k ′ ( 0 ) = δ k , k ′ c k ′ ( 1 ) = − i ℏ ∫ 0 t d t ′ ⟨ k ′ | H 1 ( t ′ ) | k ⟩ e − i ( E k − E k ′ ) t ′ / ℏ The probability of finding | k ′ ⟩ is found by evaluating | c k ′ ( t ) | 2 .
In case of constant perturbation, c k ′ ( 1 ) is calculated by
c k ′ ( 1 ) = ⟨ k ′ | H 1 | k ⟩ E k ′ − E k ( 1 − e i ( E k ′ − E k ) t / ℏ ) | c k ′ ( t ) | 2 = | ⟨ k ′ | H 1 | k ⟩ | 2 s i n 2 ( E k ′ − E k 2 ℏ t ) ( E k ′ − E k 2 ℏ ) 2 1 ℏ 2 Using the equation which is
lim α → ∞ 1 π s i n 2 ( α x ) α x 2 = δ ( x ) The transition rate of an electron from the initial state k to final state k ′ is given by
P ( k , k ′ ) = 2 π ℏ | ⟨ k ′ | H 1 | k ⟩ | 2 δ ( E k ′ − E k ) where E k and E k ′ are the energies of the initial and final states including the perturbation state and ensures the δ -function indicate energy conservation.
The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by
w ( k ) = ∑ k ′ P ( k , k ′ ) = 2 π ℏ ∑ k ′ | ⟨ k ′ | H 1 | k ⟩ | 2 δ ( E k ′ − E k ) The integral form is
w ( k ) = 2 π ℏ L 3 ( 2 π ) 3 ∫ d 3 k ′ | ⟨ k ′ | H 1 | k ⟩ | 2 δ ( E k ′ − E k ) 