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Separation of variables for quantum harmonic oscillator

  1. Oct 30, 2009 #1
    a) Show that the Hamiltonian for the quantum harmonic oscillator in 3D is separable, b) calculate the energy levels.


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    a) If it's separable H = H_x + H_y + H_z, so do I just re-arrange the kinetic and potential terms of the Hamiltonian in this case? that seems kind of trivial, as if I'm probably missing something...

    b) I assume that if the Hamiltonian is separable, we can use the method of separation of variables to find the energy, i.e., Psi(x,y,z) = X(x)Y(y)Z(z), substitute and then divide by it to find the three energies. So I get, for example,

    (1/(X(x))*(-hbar^2 / 2m d^2 / dx^2 + 1/2 k_x x^2)X

    How does that correspond to the energy level of the oscillator for the x coordinate, i.e.

    hbar*omega (n + 1/2) where n is an integer
     
    Last edited: Oct 30, 2009
  2. jcsd
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