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.

The restoring force has the form F* _{x}* = –

*k*

_{x}*x*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

_{n}*term indicates a hermite polynomial, which looks like this:*

_{x}H

_{0}(*x*) = 1H

_{1}(*x*) = 2*x*H

_{2}(*x*) = 4*x*^{2}– 2H

_{3}(*x*) = 8*x*^{3}– 12*x*H

_{4}(*x*) = 16*x*^{4}– 48*x*^{2}+ 12H

_{5}(*x*) = 32*x*^{5}– 160*x*^{3}+ 120*x*

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, E_{112} = E_{121} = E_{211}. In fact, it's possible to have more than threefold degeneracy for a 3D isotropic harmonic oscillator — for example, E_{200} = E_{020} = E_{002} = E_{110} = E_{101} = E_{011}.

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

where *n* = *n** _{x}* +

*n*

*+*

_{y}*n*

*.*

_{z}