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Section 13.4: Exploring the Two-dimensional Harmonic Oscillator
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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.
- Change the state from nx = ny =0 to nx = 0 and ny = 5. Describe the shape of the energy eigenfunction.
- 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?
- Change the state to nx = 5 and ny = 5. Describe the shape of the energy eigenfunction.
- 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.