Chapter 7, Problem 7.40P

(a)

To determine

Show that for the two fold degeneracy studied the first order correction to the wave function in degenerate perturbation theory is ${\psi }^1={\displaystyle \sum _{m\ne a,b}\frac{\alpha V_{ma}+\beta V_{mb}}{E^0-E_{nt}^0}}{\psi }_n^0$

(b)

To determine

Show that $\alpha$ and $\beta$ are determined by finding the eigenvectors of the matric where, ${\left[{\text{W}}^2\right]}_{ij}={\displaystyle \sum _{n\ne a,b}\frac{\langle {\psi }_i^0\left|H^{\prime }\right|{\psi }_m^0\rangle \langle {\psi }_m^0\left|H^{\prime }\right|{\psi }_j^0\rangle }{E^0-E_m^0}}$ and that the Eigen vectors of this matrix correspond to the second order energies $E^2$.

(c)

To determine

Show that second order degenerate perturbation theory, developed in (b) gives the correct energies to second order for the three state Hamiltonian.