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
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.