Chapter 7, Problem 7.35P

(a)

To determine

The projection operator $P_D$ and then construct the matrices ${\tilde{H}}^0$ and $\tilde{H}\text{'}$.

(b)

To determine

The eigenstates of ${\tilde{H}}^0$ and verify whether its spectrum is nondegenerate and the nondegenerate eigenstate of $H^0$, $|{\psi }_c^0\rangle =\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$ is also an eigenstate of ${\tilde{H}}^0$ with the same eigenvalue.

(c)

To determine

The “good” state and their energies to first order in the perturbation.