The interaction picture
Define the unperturbed Hamiltonian by , the time dependent perturbing Hamiltonian by and total Hamiltonian by .
The eigenstates of the unperturbed Hamiltonian are assumed to be
In the interaction picture, the state ket is defined by
By a Schrödinger equation, we see
which is a Schrödinger-like equation with the total replaced by .
Solving the differential equation, we can find the coefficient of n-state.
where, the zeroth-order term and first-order term are
The transition rate
The probability of finding is found by evaluating .
In case of constant perturbation, is calculated by
Using the equation which is
The transition rate of an electron from the initial state to final state is given by
where and are the energies of the initial and final states including the perturbation state and ensures the -function indicate energy conservation.
The scattering rate
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
The integral form is
- C. Hamaguchi (2001). Basic Semiconductor Physics. Springer. pp. 196–253.
- J.J. Sakurai. Modern Quantum Mechanics. Addison Wesley Longman. pp. 316–319.