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Section 13.4: Exploring the Two-dimensional Harmonic Oscillator

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Energy = in units of ground-state energy of 1-d well

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In Section 12.2 we considered the one-dimensional harmonic oscillator. Here we extend that result to two dimensions: V(x,y) = 1/2 mω2(x2 + y2) (even though in general ωx is not necessarily ωy).

In the animations, the energy eigenfunctions (three-dimensional plot and contour plot) and probability densities (three-dimensional plot and contour plot) for a two-dimensional quantum harmonic oscillator are shown. The animation uses ħ = 2m = 1 and ω = 2.  Since we have chosen ω = 2 and ħ = 2m = 1, the energy spectrum for each dimension is just En = (2n + 1) where n = 0, 1, 2,….  Hence, Enx ny = 2(nx + ny) + 2 where nx = 0, 1, 2,… and ny = 0, 1, 2,…. Use the sliders to change the state.

  1. Change the state from nx = ny =0 to nx = 0 and ny = 5.  Describe the shape of the energy eigenfunction.
  2. Change the state to nx = 5 and ny = 0.  Describe the shape of the energy eigenfunction. How does this energy eigenfunction's shape relate to the previous energy eigenfunction's shape?
  3. Change the state to nx = 5  and ny = 5.  Describe the shape of the energy eigenfunction.
  4. Describe the energy degeneracy of this system.

Do your results make sense?  Try to be as complete as possible and refer back to the one-dimensional solutions.

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