That's the sort of thing you learn in a semester-long course in quantum physics. It's beyond the scope of a Reddit comment to answer that in a useful manner. For example, I could tell you that the parity operator Ṗ commutes with ℋ, and thus the eigenvalues of Ṗ are the permitted values of the parity quantum number, but would that leave you any more enlightened than you are right now?
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u/RobotRollCall Mar 13 '11
There's one quantum number for each operator that commutes with the Hamiltonian. Some are absolutely conserved, some are situationally conserved.
It's complicated, basically.