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# Section 13.4: Exploring the Two-dimensional Harmonic Oscillator

Animation 1: Energy Eigenfunction | Animation 2: Energy Eigenfunction Contours | Animation 3: Probability Density | Animation 4: Probability Density Contours

Please wait for the animation to completely load.

In Section 12.2 we considered the one-dimensional harmonic oscillator. Here we extend that result to two dimensions: *V*(*x*,*y*) = 1/2 *m*ω^{2}(*x*^{2} + *y*^{2}) (even though in general ω_{x} is not necessarily ω_{y}).

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

*E*

_{n }= (2

*n*+ 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.

- Change the state from
*n*_{x }=*n*_{y }=0 to*n*_{x}= 0 and*n*_{y}= 5. Describe the shape of the energy eigenfunction. - Change the state to
*n*_{x }= 5 and*n*_{y }= 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
*n*_{x }= 5 and*n*_{y }= 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.

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