Applying the Radial Equation Inside the Square Well - dummies

# Applying the Radial Equation Inside the Square Well

In quantum physics, you can apply the radial equation inside a square well (where the radius is greater than zero and less than a). For a spherical square well potential, here’s what the radial equation looks like for the region 0 < r < a:

In this region, V(r) = –V0, so you have

Taking the V0 term over to the right gives you the following:

And here’s what dividing by r gives you:

Then, multiplying by

you get

Now make the change of variable

Using this substitution means that

This is the spherical Bessel equation. This time,

That makes sense, because now the particle is trapped in the square well, so its total energy is E + V0, not just E.

The solution to the preceding equation is a combination of the spherical Bessel functions

and the spherical Neumann functions

You can apply the same constraint here that you apply for a free particle: The wave function must be finite everywhere.

the Bessel functions look like this:

the Neumann functions reduce to

So the Neumann functions diverge for small

which makes them unacceptable for wave functions here. That means that the radial part of the wave function is just made up of spherical Bessel functions, where Al is a constant:

The whole wave function inside the square well,

is a product of radial and angular parts, and it looks like this:

are the spherical harmonics.