Quantum Physics For Dummies
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When you want to find where an electron is at any given time in a hydrogen atom, what you’re actually doing is finding how far the electron is from the proton. You can find the expectation value of r, that is, , to tell you its location. Given that the wave function is

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the following expression represents the probability that the electron will be found in the spatial element d3r:

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In spherical coordinates,

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So you can write

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as

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The probability that the electron is in a spherical shell of radius r to r + dr is therefore

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And because

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this equation becomes the following:

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The preceding equation is equal to

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(Remember that the asterisk symbol [*] means the complex conjugate. A complex conjugate flips the sign connecting the real and imaginary parts of a complex number.)

Spherical harmonics are normalized, so this just becomes

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Okay, that’s the probability that the electron is inside the spherical shell from r to r + dr. So the expectation value of r, which is , is

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which is

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This is where things get more complex, because Rnl(r) involves the Laguerre polynomials. But after a lot of math, here’s what you get:

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where r0 is the Bohr radius:

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The Bohr radius is about

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so the expectation value of the electron’s distance from the proton is

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So, for example, in the 1s state

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the expectation value of r is equal to

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And in the 4p state

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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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