Finding the Limits for Small and Large Rho of a Free Particle

In quantum physics, you can find the limits for small and large rho of a free particle. According to the spherical Bessel equation, the radial part of the wave function for a free particle looks like this:

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Take a look at the spherical Bessel functions and Neumann functions for

image1.png

Note that the Neumann functions diverge for

image2.png

Therefore, any wave function that includes the Neumann functions also diverges, which is unphysical. So the Neumann functions aren't acceptable functions in the wave function.

That means the wave function

image3.png

which equals

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equals the following:

image5.png

where

image6.png

Note that because k can take any value, the energy levels are continuous.

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