Scattering rate

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A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material.

The interaction picture[edit]

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[edit]

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[edit]

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.