Quantum Physics For Dummies
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In quantum physics, you can determine the angular part of a wave function when you work on problems that have a central potential. With central potential problems, you're able to separate the wave function into an angular part, which is a spherical harmonic, and a radial part (which depends on the form of the potential).

Central potentials are spherically symmetrical potentials, of the kind where V(r) = V(r). In other words, the potential is independent of the vector nature of the radius vector; the potential depends on only the magnitude of vector r (which is r), not on the angle of r.

So, when you have a central potential, what can you say about the angular part of


The angular part must be an eigenfunction of L2, and the eigenfunctions of L2 are the spherical harmonics,


(where l is the total angular momentum quantum number and m is the z component of the angular momentum's quantum number). The spherical harmonics equal


Here are the first several normalized spherical harmonics:


That's what the angular part of the wave function is going to be: a spherical harmonic.

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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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