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 _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_l? You find those constants through continuity constraints: At the inside/outside boundary, where r = a, the wave function and its first derivative must be continuous. So to determine A_l and B_l, you have to solve these two equations: