In quantum physics, if you know the boundary conditions of a square well, you can find theenergy levels of an electron.
The equation
![image0.png](https://www.dummies.com/wp-content/uploads/397792.image0.png)
tells you that you have to use the boundary conditions to find the constants A and B. What are the boundary conditions? The wave function must disappear at the boundaries of an infinite square well, so
![image1.png](https://www.dummies.com/wp-content/uploads/397793.image1.png)
The fact that
![image2.png](https://www.dummies.com/wp-content/uploads/397794.image2.png)
tells you right away that B must be zero because cos(0) = 1. And the fact that
![image3.png](https://www.dummies.com/wp-content/uploads/397795.image3.png)
tells you that
![image4.png](https://www.dummies.com/wp-content/uploads/397796.image4.png)
Because sine is zero when its argument is a multiple of
![image5.png](https://www.dummies.com/wp-content/uploads/397797.image5.png)
this means that
![image6.png](https://www.dummies.com/wp-content/uploads/397798.image6.png)
Note that although n = 0 is technically a solution, it yields
![image7.png](https://www.dummies.com/wp-content/uploads/397799.image7.png)
for all x, which is not normalizable, so it’s not a physical solution — the physical solutions begin with n = 1.
This equation can also be written as
![image8.png](https://www.dummies.com/wp-content/uploads/397800.image8.png)
And because
![image9.png](https://www.dummies.com/wp-content/uploads/397801.image9.png)
you have the following equation, where n = 1, 2, 3, ... — those are the allowed energy states. These are quantized states, corresponding to the quantum numbers 1, 2, 3, and so on:
![image10.png](https://www.dummies.com/wp-content/uploads/397802.image10.png)
Note that the first physical state corresponds to n = 1, which gives you this next equation:
![image11.png](https://www.dummies.com/wp-content/uploads/397803.image11.png)
This is the lowest physical state that the particles can occupy. Just for kicks, put some numbers into this, assuming that you have an electron, mass
![image12.png](https://www.dummies.com/wp-content/uploads/397804.image12.png)
confined to an infinite square well of width of the order of the Bohr radius (the average radius of an electron’s orbit in a hydrogen atom); let’s say
![image13.png](https://www.dummies.com/wp-content/uploads/397805.image13.png)
gives you this energy for the ground state:
![image14.png](https://www.dummies.com/wp-content/uploads/397806.image14.png)
That’s a very small amount, about 4 electron volts (eV — the amount of energy one electron gains falling through 1 volt). Even so, it’s already on the order of the energy of the ground state of an electron in the ground state of a hydrogen atom (13.6 eV), so you can say you’re certainly in the right quantum physics ballpark now.