Quantum Physics For Dummies, Revised Edition
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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

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Note that the Neumann functions diverge for

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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

image4.png

equals the following:

image5.png

where

image6.png

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

About This Article

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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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