The Stark effect for the 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 .
The perturbation due to an electric field in the z direction is
So our first order degenerate state perturbation theory equation is
Because of the exact degeneracy , and can be eliminated from the equation.
Lets define the one remaining nonzero (real) matrix element to be
What remains is to compute . Recall and .
If the states are not exactly degenerate, we have to leave in the diagonal terms of
Assume that the energies of the two (mixed) states are
comes from some other perturbation, like the hydrogen fine structure.
are still not mixed by the electric field.)
Jim Branson 2013-04-22