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Homework Help: Coupled Quantum Harmonic Oscillator

  1. May 1, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to transform the Hamiltonian of a coupled Harmonic Oscillator into the sum of two decoupled Hamiltonians (non-interacting oscillators).

    2. Relevant equations
    H = H1 + H2 + qxy, where H1=0.5*m*omega^2*x^2+0.5m^-1P_x^2 and H2=0.5*m*omega^2*y^2+0.5m^-1P_y^2, and q is the coupling constant

    3. The attempt at a solution
    I have tried a number of variable transformations, etc as well as attempted to complete the square to deive the proper variables that will allow me to rewrite the Hamiltonian properly. All without success. I thought completing the square and making the proper substitions would work but I still end up with a coupled term. Once this step is complete solviing the rest of the problem should be pretty straightforward.

    Any help that gets me started would be greatly appreciated.
  2. jcsd
  3. May 1, 2012 #2


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    To get rid of the cross term, you don't want to complete the square; you need to use a rotation of the coordinates.
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