In this problem you will develop an alternative approach to degenerate perturbation theory. Consider an unperturbed Hamiltonian H 0 with two degenerate states 0

In this problem you will develop an alternative approach to degenerate perturbation theory. Consider an unperturbed Hamiltonian H⁰ with two degenerate states Ψ⁰_a and Ψ⁰_b (energy E⁰), and a perturbation H'. Define the operator that projects onto the degenerate subspace:

The Hamiltonian can be written

where

The idea is to treat H̅⁰ as the “unperturbed” Hamiltonian and H̅' as the perturbation; as you’ll soon discover, H̅⁰ is nondegenerate, so we can use ordinary nondegenerate perturbation theory.
(a) First we need to find the eigenstates of H̅₀.
i. Show that any eigenstate Ψ⁰_n (other than Ψ⁰_a or Ψ⁰_b) of  H⁰ is also an eigenstate of H̅⁰ with the same eigenvalue.
ii. Show that the “good” states Ψ⁰ = aΨ⁰_a + β Ψ⁰_b (with α and β determined by solving Equation 7.30) are eigenstates of  H̅⁰ with energies E⁰ + E¹₊.
(b) Assuming that E¹₊ and E¹_- are distinct, you now have a nondegenerate unperturbed Hamiltonian H̅⁰ and you can do nondegenerate perturbation theory using the perturbation H̅. Find an expression for the energy to second order for the states Ψ⁰_± in (ii).
Comment: One advantage of this approach is that it also handles the case where the unperturbed energies are not exactly equal, but very close: E⁰_a ≈ E⁰_b. In this case one must still use degenerate perturbation theory; an important example of this occurs in the nearly-free electron approximation for calculating band structure.

Po-(+). = (7.100)