I'm wondering how one solves for the quantum harmonic oscillator using matrix methods exclusively. Given the Hamiltonian(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \hat{H} = \frac {1}{2m} \hat {P}^2 + \frac {1}{2} m \omega ^2 \hat {X}^2 [/tex]

and commutator relation

[tex] [ \hat {X} , \hat {P} ] = i \hbar \hat {I} [/tex]

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

[tex] \hat {H} \psi = E \psi [/tex]

but that is called the Schrödinger equation after all, so it seems like cheating.

Thanks.

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# Necessary axioms to derive solution to QHO problem

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