Infinite square well, in which the walls go to infinity, is a favorite problem in quantum physics. To solve for the wave function of a particle trapped in an infinite square well, you can simply solve the Schrödinger equation.
Take a look at the infinite square well in the figure.
![A square well.](https://www.dummies.com/wp-content/uploads/394817.image0.jpg)
Here’s what that square well looks like:
![image1.png](https://www.dummies.com/wp-content/uploads/394818.image1.png)
The Schrödinger equation looks like this in three dimensions:
![image2.png](https://www.dummies.com/wp-content/uploads/394819.image2.png)
Writing out the Schrödinger equation gives you the following:
![image3.png](https://www.dummies.com/wp-content/uploads/394820.image3.png)
You’re interested in only one dimension — x (distance) — in this instance, so the Schrödinger equation looks like
![image4.png](https://www.dummies.com/wp-content/uploads/394821.image4.png)
Because V(x) = 0 inside the well, the equation becomes
![image5.png](https://www.dummies.com/wp-content/uploads/394822.image5.png)
And in problems of this sort, the equation is usually written as
![image6.png](https://www.dummies.com/wp-content/uploads/394823.image6.png)
So now you have a second-order differential equation to solve for the wave function of a particle trapped in an infinite square well.
You get two independent solutions because this equation is a second-order differential equation:
![image7.png](https://www.dummies.com/wp-content/uploads/394824.image7.png)
A and B are constants that are yet to be determined.
The general solution of
![image8.png](https://www.dummies.com/wp-content/uploads/394825.image8.png)
is the sum of
![image9.png](https://www.dummies.com/wp-content/uploads/394826.image9.png)