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#1
11-29-2007, 08:37 PM
 lifes3ps Senior Member Join Date: Oct 2006 Posts: 182

can anyone help me derive the expected value of the electron radius in the form:

&lt;r&gt; =a/2*(3n^2-l(l+1))

either:
explicit form for radial wave function w/ laguerre poly's or using the reduced wave eqn.
or point me to a place/text where i can go over the derivation myself.

thanks
#2
11-29-2007, 10:35 PM
 gumpzilla Senior Member Join Date: Feb 2005 Posts: 7,911

Expected value of the electron radius in what? Looking at your answer, I'm guessing it will be the hydrogen atom. If so, ANY book on QM will have the solution to this. Griffiths is a good introductory one. Hell, Wikipedia will have the form of the radial wave function. It's Bessel functions of some flavor, I think, though I can't remember how angular momentum comes in, and you'll probably want to look up the integral anyway.
#3
11-30-2007, 11:23 PM
 TWCReborn Junior Member Join Date: Nov 2007 Location: Los Angeles Posts: 16

Anyhow you are not looking for the electron's radius. Electron's don't have a radius. Classical models give electrons a radius. Seeing that you have n and l in the equation, I presume you are talking about the &lt;r&gt; (expectation value) of a hydrogen atom? The wavefunction in question, is probably the wavefunctions for the hydrogen atom (Ym,lR). Laguerre is in any introductory QM book if that is what you are looking for. You would simply sandwhich r between your wavefunction and its conjugate and integrate. The setup should be immediately obvious once you look through relevant equations. The rest is fiddling with the math (calculus tricks).

The integrals in these types of problems can sometimes be simplified by considering their parity (odd/even). I don't think this is one of those. Anyhow, beyond setting up the integral, the rest is calculus, so asking someone for an answer is like asking them to do an integral.

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