# How to Find the Energy Level of a Harmonic Oscillator: An Example

Your quantum physics instructor may ask you to find the energy level of a harmonic oscillator. The best way to learn how is through an example. Say that you have a proton undergoing harmonic oscillation with

as shown in the figure.

A proton undergoing harmonic oscillation.

What are the energies of the various energy levels of the proton? You know that in general,

So here are the energies of the proton, in megaelectron volts (MeV):

And so on.

Now what about the wave functions? The general form of

Convert all length measurements into femtometers

where *x* is measured in femtometers:

Here are a couple more wave functions: