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Applying the Radial Equation Inside the Square Well

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Updated:  
2016-03-26 14:05:24
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From The Book:  
String Theory For Dummies
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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.

About This Article

This article is from the book: 

About the book author:

Dr. Steven Holzner has written more than 40 books about physics and programming. He was a contributing editor at PC Magazine and was on the faculty at both MIT and Cornell. He has authored Dummies titles including Physics For Dummies and Physics Essentials For Dummies. Dr. Holzner received his PhD at Cornell.