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).