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](https://www.dummies.com/wp-content/uploads/396245.image0.png)
looks like, you can determine the first excited state,
![image1.png](https://www.dummies.com/wp-content/uploads/396246.image1.png)
Say you’re given this as your starting point:
![image2.png](https://www.dummies.com/wp-content/uploads/396247.image2.png)
And you know that
![image3.png](https://www.dummies.com/wp-content/uploads/396248.image3.png)
is the following:
![image4.png](https://www.dummies.com/wp-content/uploads/396249.image4.png)
And because
![image5.png](https://www.dummies.com/wp-content/uploads/396250.image5.png)
you get the following equation:
![image6.png](https://www.dummies.com/wp-content/uploads/396251.image6.png)
Given that the wave function for the ground state of a quantum mechanical harmonic oscillator is
![image7.png](https://www.dummies.com/wp-content/uploads/396252.image7.png)
So what does
![image8.png](https://www.dummies.com/wp-content/uploads/396253.image8.png)
look like?
![The first excited state of a quantum mechanical harmonic oscillator.](https://www.dummies.com/wp-content/uploads/396254.image9.jpg)
The first excited state of a quantum mechanical harmonic oscillator.
You can see a graph of
![image10.png](https://www.dummies.com/wp-content/uploads/396255.image10.png)
in the figure, where it has one node (transition through the x axis).