Determining the Angular Part of a Wave Function
Applying the Spherical Bessel and Neumann Functions to a Free Particle
How to Work with a Cubic Potential

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:

image0.png

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

image1.png

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

image2.png

And here's what dividing by r gives you:

image3.png

Then, multiplying by

image4.png

you get

image5.png

Now make the change of variable

image6.png

Using this substitution means that

image7.png

This is the spherical Bessel equation. This time,

image8.png

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

image9.png

and the spherical Neumann functions

image10.png

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

image11.png

the Bessel functions look like this:

image12.png

the Neumann functions reduce to

image13.png

So the Neumann functions diverge for small

image14.png

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:

image15.png

The whole wave function inside the square well,

image16.png

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

image17.png

are the spherical harmonics.

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