Can't Prove formula from Sakurai's Modern QM @ Perturbation Theory I am studying Perturbation Theory from J.J. Sakurai's textbook Modern Quantum Mechanics. I am having trouble proving formulas on page 299 (5.2.5) and (5.2.6) from the previous ones [mainly (5.2.4)]. Can anyone help?
This is a problem of working with perturbation theory when the non-perturbed state has degeneracy. To elaborate I do not get why $(E-H_0-\lambda P_1VP_1)$ is not singular.
More importantly is the equation below correct and if so why:
$$(E-\lambda P_1VP_1)|l \rangle = E_D^{(0)}|l \rangle $$
That is the core of my problem. If the above equation - in the framework of Sakurai's statement of the problem - is correct I can work out all the rest on my own. However I do not get why this equation would hold true.
 A: I think there's a typo. In the line after (5.2.4), the expression
$$P_1(E - H_0 - \lambda P_1 V P_1)$$
should have the second parenthesis moved to read
$$P_1(E - H_0 - \lambda P_1 V) P_1$$
which is equivalent to
$$E P_1 - P_1 H_0 P_1 - \lambda P_1 V P_1$$
Saying that this is not singular is the same as saying it doesn't map anything to zero. It doesn't, for the reason the text says: $E$ and the eigenvalues of $P_1 H_0 P_1$ cannot be the same since $E$ is just the perturbed value of $E_0$ while $H_0$ is acting on states that are orthogonal to the subspace $\{|m^{(0)}\rangle\}$.
A: The equation as stated by you $E- \lambda P_1 V P_1 |l\rangle = E^{(0)}_D |l\rangle $ seems correct to me. This is so because $\lambda$ is small. Hence, $E$ is a small departure from the unperturbed energies in the degenerate subspace, i.e. $E_D^{(0)}$. So in the limit $\lambda \rightarrow 0$, the following two limits are approached: $E \rightarrow$ $E_D^{(0)}$ and $\lambda P_1 V P_1 \rightarrow 0$.
BTW @Mark Eichenlaub, I don't think there is any typo in the book (as suggested by you), primarily because I did not find any wrong statement/equations in the book. If you found out any, please point it out.
