Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

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

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Necessary axioms to derive solution to QHO problem
Loading...