A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material.
Define the unperturbed Hamiltonian by
, the time dependent perturbing 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
Solving the differential equation, we can find the coefficient of n-state.
where, the zeroth-order term and first-order term are The probability of finding
In case of constant perturbation,
is calculated by Using the equation which is The transition rate of an electron from the initial state
are the energies of the initial and final states including the perturbation state and ensures the
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