In quantum physics, you can find the wave function of the ground state of a quantum oscillator, such as the one shown in the figure, which takes the shape of a gaussian curve.
![The ground state of a quantum mechanical harmonic oscillator.](https://www.dummies.com/wp-content/uploads/395332.image0.jpg)
The ground state of a quantum mechanical harmonic oscillator.
As a gaussian curve, the ground state of a quantum oscillator is
![image1.png](https://www.dummies.com/wp-content/uploads/395333.image1.png)
How can you figure out A? Wave functions must be normalized, so the following has to be true:
![image2.png](https://www.dummies.com/wp-content/uploads/395334.image2.png)
Substituting for
![image3.png](https://www.dummies.com/wp-content/uploads/395335.image3.png)
gives you this next equation:
![image4.png](https://www.dummies.com/wp-content/uploads/395336.image4.png)
You can evaluate this integral to be
![image5.png](https://www.dummies.com/wp-content/uploads/395337.image5.png)
Therefore,
![image6.png](https://www.dummies.com/wp-content/uploads/395338.image6.png)
This means that the wave function for the ground state of a quantum mechanical harmonic oscillator is
![image7.png](https://www.dummies.com/wp-content/uploads/395339.image7.png)
Cool. Now you’ve got an exact wave function.