In this problem you will develop an alternative approach to degenerate perturbation theory. Consider an unperturbed Hamiltonian H0 with two degenerate states ψa0 and ψb0 (energy E0), and a perturbation H′. Define the operator that projects onto the degenerate subspace:
PD=∣ψa0⟩⟨ψa0∣+∣ψb0⟩⟨ψb0∣.
The Hamiltonian can be written
H=H0+H′=H~0+H~′
where
H~0=H0+PDH′PD,H~′=H′−PDH′PD.
The idea is to treat H~0 as the "unperturbed" Hamiltonian and H~′ as the perturbation; as you'll soon discover, H~0 is nondegenerate, so we can use ordinary nondegenerate perturbation theory.
(a) First we need to find the eigenstates of H~0.
i. Show that any eigenstate ψn0 (other than ψa0 or ψb0) of H0 is also an eigenstate of H~0 with the same eigenvalue.
ii. Show that the "good" states ψ0=αψa0+βψb0 (with α and β determined by solving Equation 7.30) are eigenstates of H~0 with energies E0+E±1.
(b) Assuming that E+1 and E−1 are distinct, you now have a nondegenerate unperturbed Hamiltonian H~0 and you can do nondegenerate perturbation theory using the perturbation H~′. Find an expression for the energy to second order for the states ψ±0 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: Ea0≈Eb0. 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.