# 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:

1. 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 $\lambda \rightarrow 0$.
2. 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).

• You're much more likely to get some good answers if you write the question in a way that doesn't require someone to go find a copy of the book you're using.
– Chris
Oct 21, 2017 at 4:35
• Anyone still looking into this procedure or trying to calculate eigenkets for the degenerate case, might find this note helpful: arxiv.org/pdf/2002.05001.pdf Nov 17, 2021 at 0:03