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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

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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,

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

image3.png

And so on.

Now what about the wave functions? The general form of

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Convert all length measurements into femtometers

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where x is measured in femtometers:

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Here are a couple more wave functions:

image7.png
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