# 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

looks like, you can determine the first excited state,

Say you’re given this as your starting point:

And you know that

is the following:

And because

you get the following equation:

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

So what does

look like?

The first excited state of a quantum mechanical harmonic oscillator.

You can see a graph of

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