Degenerate perturbation theory: Sakurai's statement does not make sense Book: Modern Quantum Mechanics (Revised edition): J J Sakurai
After Eq. 5.2.15, Sakurai summarizes the recipe to treat degenerate perturbation theory in four bullet points. The last two bullet points are:


*Identify the roots of the secular equation with the first-order energy shifts; the base kets that diagonalize the $V$ matrix are the correct zeroth-order kets that the perturbed kets approach in the limit $\lambda \rightarrow 0$.

*For higher orders, use the formulas of the corresponding nondegenerate
perturbation theory except in the summations, where we exclude all contributions from the unperturbed kets in the degenerate subspace $D$.


Using point (3) one can say that up to the zeroth order the kets of the degenerate subspace are the ones which diagonalize the perturbation matrix. Let's move on to seek the first order correction to these kets. Point (4) above suggests using the non-degenerate results which I again reproduce (Sakurai Eq. 5.1.44):
$
|n\rangle = |n^{(0)} \rangle + \lambda \sum_{k \neq n} |k^{(0)}\rangle  \frac{V_{kn}}{ E_n^{(0)}- E_k^{(0)}} + \cal{O}(\lambda^2).
$
For those who don't have the book: $|n^{(0)} \rangle$ = unperturbed ket, $|n\rangle$ = perturbed ket, $E_n^{(0)}$= energy of the unperturbed $n$th ket, $\lambda$= perturbation parameter (0<$\lambda$<1) and $V_{kn}= k,n$ matrix element of the perturbation $V$.
So to me, Sakurai seems to suggest to use the above non-degenerate formula to find the first order correction (of the order $\lambda$) for the perturbed ket, where $k$ does not run over the unperturbed kets of the degenerate subspace. Let's call this 'method no. 1'.
Confusion: After Eq. 5.2.14, Sakurai says "If we add together (5.2.6) and (5.2. 14), we get the eigenvector accurate to order $\lambda.$". Let's call this 'method no 2'. Now, I totally agree with method no. 2. But I don't see how does method no. 1 give the same result as method no. 2. For example, method no. 2 says the solution is 5.2.6 + 5.2.14, whereas one can clearly see that 5.2.6 agrees perfectly well with method no. 1. But unfortunately Sakurai's method no. 2 is not just 5.2.6, but (5.2.6+5.2.14), and hence methods 1 and 2 don't agree. What troubles me further is that many other books make similar statements as Sakurai's bullet point 4 above (e.g. Griffiths, Ballentine, Liboff, Auletta).
 A: After some work, I have reached the following conclusion about this issue. There indeed seems a contradiction between 'method 1' and 'method 2' as defined in the question. I think 'method 2' is the right way to proceed. Here are the reasons:


*

*When Sakurai advocates method 2, he does it after having derived it. Hence we can trust it, whereas the part of Sakurai which seems to suggest method 1 as the right approach ('bullet 4' as stated in the question above), has been given without any proof/derivation. 

*This is also true for the other books which suggest method 2 (e.g. Griffiths, Ballentine, Liboff, Auletta). None of them show explicitly why method 2 is correct. They just state it.

*The revered book 'Methods of Mathematical Physics' by Richard Courant and D Hilbert suggest method 2 on pg 348.

*By the way, method 2 still seems correct, but only for the energy eigenvalue corrections, and not for the eigenvector correction. It could be that the above books (including Sakurai) implicitly mean that method 2 is only applicable for energy corrections.
