How to Find Any Excited State of a Harmonic Oscillator

If you can determine the wave function for the ground state of a quantum mechanical harmonic oscillator, then you can find any excited state of that harmonic oscillator.

So, if you know what

image0.png

looks like, you can determine the first excited state,

image1.png

Say you’re given this as your starting point:

image2.png

And you know that

image3.png

is the following:

image4.png

And because

image5.png

you get the following equation:

image6.png

Given that the wave function for the ground state of a quantum mechanical harmonic oscillator is

image7.png

So what does

image8.png

look like?

The first excited state of a quantum mechanical harmonic oscillator.
The first excited state of a quantum mechanical harmonic oscillator.

You can see a graph of

image10.png

in the figure, where it has one node (transition through the x axis).

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