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What is a good resource to learn about higher degree degenerate perturbation theory - one that involves mathematics that isn't much more advanced than first order perturbation theory? I've looked around and I've only found Sakurai talk about it but he uses projections operators and other fancy mathematics. Also, does anyone have any examples of it being used?

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I believe griffith's "Introduction to QM" also provides a introduction to higher order perturbations (well actually most books on QM do).

But you will always encounter projections ! This is because of the fact that for the second order perturbation in the energy, you'll need the first order perturbation on your wavefunction (and for the n-th order in the energy the (n-1)-th order in your wavefunction).

So I'm afraid that you're stuck with projections of wavefunctions in your Hilberspace. Sarukai is a great reference and I'd really recommend that one to look for the aspects of perturbation theory. Try to do the calculations yourself and write in each step the logic of that specific step, that will help a lot !

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You can also have a look in Landau and Lifshitz (Quantum Mechanics - Non-relativistic Theory, where in §39. The secular equation, degenerate perturbation theory is treated, then there is specifically to your question

Problem 2.: "Derive the formulae for the correction to the eigenfunctions in the first approximation and to the eigenvalues in the second approximation."

Landau's treatment is usually a little different from others', and thus might help to gain more insight.

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