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Necessary axioms to derive solution to QHO problem

  1. Apr 14, 2014 #1
    I'm wondering how one solves for the quantum harmonic oscillator using matrix methods exclusively. Given the Hamiltonian
    [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.

    Last edited: Apr 14, 2014
  2. jcsd
  3. Apr 14, 2014 #2


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    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.
  4. Apr 14, 2014 #3
    Very interesting, thank you dextercioby.

    edit: Oh wait, all three matrices have to be Hermitian, don't they? That's a restriction, too.
    Last edited: Apr 14, 2014
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