Working with Three-Dimensional Harmonic Oscillators

In quantum physics, when you are working in one dimension, the general particle harmonic oscillator looks like the figure shown here, where the particle is under the influence of a restoring force — in this example, illustrated as a spring.

A harmonic oscillator.

The restoring force has the form F_x = –k_xx in one dimension, where k_x is the constant of proportionality between the force on the particle and the location of the particle. The potential energy of the particle as a function of location x is

This is also sometimes written as

Now take a look at the harmonic oscillator in three dimensions. In three dimensions, the potential looks like this:

Now that you have a form for the potential, you can start talking in terms of Schrödinger's equation:

Substituting in for the three-dimension potential, V(x, y, z), gives you this equation:

Take this dimension by dimension. Because you can separate the potential into three dimensions, you can write

Therefore, the Schrödinger equation looks like this for x:

Solving that equation, you get this next solution:

where

and n_x = 0, 1, 2, and so on. The H_nx term indicates a hermite polynomial, which looks like this:

• H₀(x) = 1

• H₁(x) = 2x

• H₂(x) = 4x² – 2

• H₃(x) = 8x³ – 12x

• H₄(x) = 16x⁴ – 48x² + 12

• H₅(x) = 32x⁵ – 160x³ + 120x

Therefore, you can write the wave function like this:

That's a relatively easy form for a wave function, and it's all made possible by the fact that you can separate the potential into three dimensions.

What about the energy of the harmonic oscillator? The energy of a one-dimensional harmonic oscillator is

And by analogy, the energy of a three-dimensional harmonic oscillator is given by

Note that if you have an isotropic harmonic oscillator, where

the energy looks like this:

As for the cubic potential, the energy of a 3D isotropic harmonic oscillator is degenerate. For example, E₁₁₂ = E₁₂₁ = E₂₁₁. In fact, it's possible to have more than threefold degeneracy for a 3D isotropic harmonic oscillator — for example, E₂₀₀ = E₀₂₀ = E₀₀₂ = E₁₁₀ = E₁₀₁ = E₀₁₁.

In general, the degeneracy of a 3D isotropic harmonic oscillator is

where n = n_x + n_y + n_z.