In quantum physics, if you are given the wave equation for a particle in an infinite square well, you may be asked to normalize the wave function. For example, start with the following wave equation:
![image0.png](https://www.dummies.com/wp-content/uploads/397809.image0.png)
The wave function is a sine wave, going to zero at x = 0 and x = a. You can see the first two wave functions plotted in the following figure.
![Wave functions in a square well.](https://www.dummies.com/wp-content/uploads/397810.image1.jpg)
Normalizing the wave function lets you solve for the unknown constant A. In a normalized function, the probability of finding the particle between
![image2.png](https://www.dummies.com/wp-content/uploads/397811.image2.png)
adds up to 1 when you integrate over the whole square well, x = 0 to x = a:
![image3.png](https://www.dummies.com/wp-content/uploads/397812.image3.png)
Substituting for
![image4.png](https://www.dummies.com/wp-content/uploads/397813.image4.png)
gives you the following:
![image5.png](https://www.dummies.com/wp-content/uploads/397814.image5.png)
Here’s what the integral in this equation equals:
![image6.png](https://www.dummies.com/wp-content/uploads/397815.image6.png)
So from the previous equation,
![image7.png](https://www.dummies.com/wp-content/uploads/397816.image7.png)
Solve for A:
![image8.png](https://www.dummies.com/wp-content/uploads/397817.image8.png)
Therefore, here’s the normalized wave equation with the value of A plugged in:
![image9.png](https://www.dummies.com/wp-content/uploads/397818.image9.png)
And that’s the normalized wave function for a particle in an infinite square well.