# Applying the Radial Equation Outside the Square Well

In quantum physics, you can apply the radial equation outside a square well (where the radius is greater than *a*). In the region *r* > *a*, the particle is just like a free particle, so here’s what the radial equation looks like:

You solve this equation as follows:

you substitute

so that R* _{nl}*(

*r*) becomes

Using this substitution means that the radial equation takes the following form:

The solution is a combination of spherical Bessel functions and spherical Neumann functions, where B* _{l}* is a constant:

If the energy E < 0, you must have C_{l}* *= i B* _{l}*“, so that the wave function decays exponentially at large distances

*r*. So the radial solution outside the square well looks like this, where

Given that the wave function inside the square well is

So how do you find the constants A* _{l}* and B

*? You find those constants through continuity constraints: At the inside/outside boundary, where*

_{l}*r*=

*a*, the wave function and its first derivative must be continuous. So to determine A

*and B*

_{l}*, you have to solve these two equations:*

_{l}