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 ω_x is not necessarily ω_y).

In the animations, the wave functions (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 E_n = (2n + 1) where n = 0, 1, 2,….  Hence, E_{nx ny} = 2(n_x + n_y) + 2 where n_x = 0, 1, 2,… and n_y = 0, 1, 2,…. Use the sliders to change the state.

1. Change the state from n_x = n_y =0 to n_x = 0 and n_y = 5.  Describe the shape of the wave function.
2. Change the state to n_x = 5 and n_y = 0.  Describe the shape of the wave function.  How does this wave function's shape relate to the previous wave function's shape?
3. the state to n_x = 5  and n_y = 5.  Describe the shape of the wave function.
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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