Assume that two or more states are (nearly) degenerate.
Define
to be the set of those nearly degenerate states.
Choose a set of basis state in
which are orthonormal

where and are in the set . We will use the indices and to label the states in .

By looking at the zeroth and first order terms in the Schrödinger equation and dotting it into one of the
degenerate states
, we
derive
the energy equation for first order (nearly) degenerate state perturbation theory

This is an eigenvalue equation with as many solutions as there are degnerate states in our set. audio

We recognize this as simply the (matrix) energy eigenvalue equation limited the list of degenerate states. We solve the equation to get the energy eigenvalues and energy eigenstates, correct to first order.

Written as a matrix, the equation is

where is the matrix element of the full Hamiltonian. If there are n nearly degenerate states, there are n solutions to this equation.

The Stark effect for the (principle quantum number) n=2 states of hydrogen requires the use of degenerate state perturbation theory since there are four states with (nearly) the same energies. For our first calculation, we will ignore the hydrogen fine structure and assume that the four states are exactly degenerate, each with unperturbed energy of . That is . The degenerate states , , , and .

* Example:
The Stark Effect for n=2 States.*

The perturbation due to an electric field in the z direction is . The linear combinations that are found to diagonalize the full Hamiltonian in the subspace of degenerate states are: , and with energies of , , and .

Jim Branson 2013-04-22