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mathingenue

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- TL;DR Summary
- Meaning of good quantum numbers and relationship to perturbation theory and matrix diagonalization

Hello folks, I am currently studying from Griffiths' Introduction to Quantum Mechanics and I've got a doubt about good quantum numbers that the text has been unable to solve.

As I understand it, good quantum numbers are the eigenvalues of the eigenvectors of an operator O that remain eigenvectors of such an operator with the same eigenvalues as time evolves. At least that's the Wikipedia definition: https://en.wikipedia.org/wiki/Good_quantum_number. This is equivalent to saying that the operator O commutes with the Hamiltonian H of the problem in question.

But what does this have to do with degenerate perturbation theory? As I see it, degeneracy introduces two problems in the use of nondegenerate perturbation theory: 1) you don't know which states to use, since any linear combination of the degenerate eigenstates is also an eigenstate with the same energy, and 2) the expressions derived in the nondegenerate case involve dividing by zero when there is degeneracy. How do good quantum numbers help solve these problems?

And what does all of this have to do with diagonalizing the Hamiltonian matrix?

Any help would be greatly appreciated, and sorry for the (very) broad questions, I couldn't find any other way to put it.

As I understand it, good quantum numbers are the eigenvalues of the eigenvectors of an operator O that remain eigenvectors of such an operator with the same eigenvalues as time evolves. At least that's the Wikipedia definition: https://en.wikipedia.org/wiki/Good_quantum_number. This is equivalent to saying that the operator O commutes with the Hamiltonian H of the problem in question.

But what does this have to do with degenerate perturbation theory? As I see it, degeneracy introduces two problems in the use of nondegenerate perturbation theory: 1) you don't know which states to use, since any linear combination of the degenerate eigenstates is also an eigenstate with the same energy, and 2) the expressions derived in the nondegenerate case involve dividing by zero when there is degeneracy. How do good quantum numbers help solve these problems?

And what does all of this have to do with diagonalizing the Hamiltonian matrix?

Any help would be greatly appreciated, and sorry for the (very) broad questions, I couldn't find any other way to put it.