Now we will work the full problem with no assumptions about which perturbation is stronger.
This is really not that hard so if we were just doing this problem on the homework, this assumption
free method would be the one to use.
The reason we work the problem all three ways is as an example of how to apply degenerate state
perturbation theory to other problems.
We continue on as in the last section but work in the states of
The matrix for
The top part is already diagonal so we only need to work in bottom
right 2 by 2 matrix, solving the eigenvalue problem.
Setting the determinant equal to zero, we get
The eigenvalues for the
states, which mix differently as a function
of the field strength, are
The eigenvalues for the other two states which remain eigenstates independent of
the field strength are