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I know that the eigenvalue of the hamiltonian applied to an eigenket is an energy level.

H |a> = E |a>

But my question is if I am given an equation for a specific Hamiltonian:

H = (something arbitrary)

And asked to find the energy levels of the object I am given that hamiltonian for (say a molecule or a particle) and no other information, what strategy do I use?

If the Hamiltonian is comprised of angular momentum operators, can I just say my object is an arbitrary eigenket = |n, l, m> and find the general eigenvalue of when my hamiltonian is applied to that?