Applying the Spherical Bessel and Neumann Functions to a Free Particle

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

image1.png

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

image8.png

and the spherical Neumann functions

image9.png

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

image13.png

Here are the first few iterations of

image14.png
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