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

Interaction picture

In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...

, the state ket is defined by

By a Schrödinger equation

Schrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

, we see
which is a Schrödinger-like equation with the total replaced by .

Solving the differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

, 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

Electron scattering

Electron scattering is the process whereby an electron is deflected from its original trajectory. As they are charged particles, they are subject to electromagnetic forces.-Phenomena:...

from an initial state k to a final state k', and is defined by


The integral form is