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

By Steven Holzner

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.

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: