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

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.

The Schrödinger equation looks like this in three dimensions, where

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is the Laplacian operator:

image1.png

And the Laplacian operator looks like this in rectangular coordinates:

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In spherical coordinates, it's a little messy, but you can simplify later. Check out the spherical Laplacian operator:

image3.png

Here, L2 is the square of the orbital angular momentum:

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So in spherical coordinates, the Schrödinger equation for a central potential looks like this when you substitute in the terms:

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Take a look at the preceding equation. The first term actually corresponds to the radial kinetic energy — that is, the kinetic energy of the particle moving in the radial direction. The second term corresponds to the rotational kinetic energy. And the third term corresponds to the potential energy.

So what can you say about the solutions to this version of the Schrödinger equation? You can note that the first term depends only on r, as does the third, and that the second term depends only on angles. So you can break the wave function,

image6.png

into two parts:

  • A radial part

  • A part that depends on the angles

This is a special property of problems with central potentials.

About This Article

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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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