In quantum physics, you can apply the spherical Bessel and Neumann functions to a free particle (a particle which is not constrained by any potential). The wave function in spherical coordinates takes this form:
![image0.png](https://www.dummies.com/wp-content/uploads/397300.image0.png)
and
![image1.png](https://www.dummies.com/wp-content/uploads/397301.image1.png)
gives you the spherical harmonics. The problem is now to solve for the radial part, Rnl(r). Here's the radial equation:
![image2.png](https://www.dummies.com/wp-content/uploads/397302.image2.png)
For a free particle, V(r) = 0, so the radial equation becomes
![image3.png](https://www.dummies.com/wp-content/uploads/397303.image3.png)
The way you usually handle this equation is to substitute
![image4.png](https://www.dummies.com/wp-content/uploads/397304.image4.png)
and because you have a version of the same equation for each n index it is convenient to simply remove it, so that Rnl (r) becomes
![image5.png](https://www.dummies.com/wp-content/uploads/397305.image5.png)
This substitution means that
![image6.png](https://www.dummies.com/wp-content/uploads/397306.image6.png)
becomes the following:
![image7.png](https://www.dummies.com/wp-content/uploads/397307.image7.png)
The radial part of the equation looks tough, but the solutions turn out to be well-known — this equation is called the spherical Bessel equation, and the solution is a combination of the spherical Bessel functions
![image8.png](https://www.dummies.com/wp-content/uploads/397308.image8.png)
and the spherical Neumann functions
![image9.png](https://www.dummies.com/wp-content/uploads/397309.image9.png)
where Al and Bl are constants. So what are the spherical Bessel functions and the spherical Neumann functions? The spherical Bessel functions are given by
![image10.png](https://www.dummies.com/wp-content/uploads/397310.image10.png)
Here's what the first few iterations of
![image11.png](https://www.dummies.com/wp-content/uploads/397311.image11.png)
look like:
![image12.png](https://www.dummies.com/wp-content/uploads/397312.image12.png)
How about the spherical Neumann functions? The spherical Neumann functions are given by
![image13.png](https://www.dummies.com/wp-content/uploads/397313.image13.png)
Here are the first few iterations of
![image14.png](https://www.dummies.com/wp-content/uploads/397314.image14.png)