In quantum physics, you can apply the radial equation inside a square well (where the radius is greater than zero and less than a). For a spherical square well potential, here's what the radial equation looks like for the region 0 < r < a:
![image0.png](https://www.dummies.com/wp-content/uploads/397326.image0.png)
In this region, V(r) = –V0, so you have
![image1.png](https://www.dummies.com/wp-content/uploads/397327.image1.png)
Taking the V0 term over to the right gives you the following:
![image2.png](https://www.dummies.com/wp-content/uploads/397328.image2.png)
And here's what dividing by r gives you:
![image3.png](https://www.dummies.com/wp-content/uploads/397329.image3.png)
Then, multiplying by
![image4.png](https://www.dummies.com/wp-content/uploads/397330.image4.png)
you get
![image5.png](https://www.dummies.com/wp-content/uploads/397331.image5.png)
Now make the change of variable
![image6.png](https://www.dummies.com/wp-content/uploads/397332.image6.png)
Using this substitution means that
![image7.png](https://www.dummies.com/wp-content/uploads/397333.image7.png)
This is the spherical Bessel equation. This time,
![image8.png](https://www.dummies.com/wp-content/uploads/397334.image8.png)
That makes sense, because now the particle is trapped in the square well, so its total energy is E + V0, not just E.
The solution to the preceding equation is a combination of the spherical Bessel functions
![image9.png](https://www.dummies.com/wp-content/uploads/397335.image9.png)
and the spherical Neumann functions
![image10.png](https://www.dummies.com/wp-content/uploads/397336.image10.png)
You can apply the same constraint here that you apply for a free particle: The wave function must be finite everywhere.
![image11.png](https://www.dummies.com/wp-content/uploads/397337.image11.png)
the Bessel functions look like this:
![image12.png](https://www.dummies.com/wp-content/uploads/397338.image12.png)
the Neumann functions reduce to
![image13.png](https://www.dummies.com/wp-content/uploads/397339.image13.png)
So the Neumann functions diverge for small
![image14.png](https://www.dummies.com/wp-content/uploads/397340.image14.png)
which makes them unacceptable for wave functions here. That means that the radial part of the wave function is just made up of spherical Bessel functions, where Al is a constant:
![image15.png](https://www.dummies.com/wp-content/uploads/397341.image15.png)
The whole wave function inside the square well,
![image16.png](https://www.dummies.com/wp-content/uploads/397342.image16.png)
is a product of radial and angular parts, and it looks like this:
![image17.png](https://www.dummies.com/wp-content/uploads/397343.image17.png)
are the spherical harmonics.