In quantum physics, you can find the limits for small and large rho of a free particle. According to the spherical Bessel equation, the radial part of the wave function for a free particle looks like this:
![image0.png](https://www.dummies.com/wp-content/uploads/397317.image0.png)
Take a look at the spherical Bessel functions and Neumann functions for
![image1.png](https://www.dummies.com/wp-content/uploads/397318.image1.png)
Note that the Neumann functions diverge for
![image2.png](https://www.dummies.com/wp-content/uploads/397319.image2.png)
Therefore, any wave function that includes the Neumann functions also diverges, which is unphysical. So the Neumann functions aren't acceptable functions in the wave function.
That means the wave function
![image3.png](https://www.dummies.com/wp-content/uploads/397320.image3.png)
which equals
![image4.png](https://www.dummies.com/wp-content/uploads/397321.image4.png)
equals the following:
![image5.png](https://www.dummies.com/wp-content/uploads/397322.image5.png)
where
![image6.png](https://www.dummies.com/wp-content/uploads/397323.image6.png)
Note that because k can take any value, the energy levels are continuous.