Necessary axioms to derive solution to QHO problem

[snoopies622]

I'm wondering how one solves for the quantum harmonic oscillator using matrix methods exclusively. Given the Hamiltonian
\hat{H} = \frac {1}{2m} \hat {P}^2 + \frac {1}{2} m \omega ^2 \hat {X}^2
and commutator relation
[ \hat {X} , \hat {P} ] = i \hbar \hat {I}
is that enough for premises? To me it looks like two equations and three unknowns. Are any other assumptions — either classical or quantum — allowed? It is tempting to introduce
\hat {H} \psi = E \psi
but that is called the Schrödinger equation after all, so it seems like cheating.

Thanks.

 

[dextercioby]

Yes, you need the extra assumption that the Hamiltonian has at least 1 eigenvector in the Hilbert space in which you represent the operator algebra. This amounts to saying the Hamiltonian is essentially self-adjoint on a representation space of the X,P,H operator algebra.

 

[snoopies622]

Very interesting, thank you dextercioby.

edit: Oh wait, all three matrices have to be Hermitian, don't they? That's a restriction, too.