In quantum physics, you can determine the radial part of a wave function when you work on problems that have a central potential. With central potential problems, you’re able to separate the wave function into a radial part (which depends on the form of the potential) and an angular part, which is a spherical harmonic.
You can give the radial part of the wave function the name Rnl(r), where n is a quantum number corresponding to the quantum state of the radial part of the wave function and l is the total angular momentum quantum number. The radial part is symmetric with respect to angles, so it can’t depend on m, the quantum number of the z component of the angular momentum. In other words, the wave function for particles in central potentials looks like the following equation in spherical coordinates:
![image0.png](https://www.dummies.com/wp-content/uploads/397037.image0.png)
The next step is to solve for Rnl(r) in general. Substituting
![image1.png](https://www.dummies.com/wp-content/uploads/397038.image1.png)
from the preceding equation into the Schrödinger equation,
![image2.png](https://www.dummies.com/wp-content/uploads/397039.image2.png)
gives you
![image3.png](https://www.dummies.com/wp-content/uploads/397040.image3.png)
Okay, what can you make of this? First, note that the spherical harmonics are eigenfunctions of L2 (that’s the whole reason for using them), with eigenvalue
![image4.png](https://www.dummies.com/wp-content/uploads/397041.image4.png)
So the last term in this equation is simply
![image5.png](https://www.dummies.com/wp-content/uploads/397042.image5.png)
That means that
![image6.png](https://www.dummies.com/wp-content/uploads/397043.image6.png)
takes the form
![image7.png](https://www.dummies.com/wp-content/uploads/397044.image7.png)
which equals
![image8.png](https://www.dummies.com/wp-content/uploads/397045.image8.png)
The preceding equation is the one you use to determine the radial part of the wave function, Rnl(r). It’s called the radial equation for a central potential.
When you solve the radial equation for Rnl(r), you can then find
![image9.png](https://www.dummies.com/wp-content/uploads/397046.image9.png)
because you already know
![image10.png](https://www.dummies.com/wp-content/uploads/397047.image10.png)
So, you’re simply finding the solution to the radial equation.