Quantum Physics For Dummies, Revised Edition
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In quantum physics, you can apply the spherical Bessel and Neumann functions to a free particle (a particle which is not constrained by any potential). The wave function in spherical coordinates takes this form:

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and

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gives you the spherical harmonics. The problem is now to solve for the radial part, Rnl(r). Here's the radial equation:

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For a free particle, V(r) = 0, so the radial equation becomes

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The way you usually handle this equation is to substitute

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and because you have a version of the same equation for each n index it is convenient to simply remove it, so that Rnl (r) becomes

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This substitution means that

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becomes the following:

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The radial part of the equation looks tough, but the solutions turn out to be well-known — this equation is called the spherical Bessel equation, and the solution is a combination of the spherical Bessel functions

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and the spherical Neumann functions

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where Al and Bl are constants. So what are the spherical Bessel functions and the spherical Neumann functions? The spherical Bessel functions are given by

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Here's what the first few iterations of

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look like:

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How about the spherical Neumann functions? The spherical Neumann functions are given by

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Here are the first few iterations of

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About This Article

This article is from the book:

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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