Two Plus Two Newer Archives  

Go Back   Two Plus Two Newer Archives > Other Topics > Science, Math, and Philosophy

Reply
 
Thread Tools Display Modes
  #1  
Old 11-29-2007, 08:37 PM
lifes3ps lifes3ps is offline
Senior Member
 
Join Date: Oct 2006
Posts: 182
Default Electron radius-quantum

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

<r> =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
Reply With Quote
  #2  
Old 11-29-2007, 10:35 PM
gumpzilla gumpzilla is offline
Senior Member
 
Join Date: Feb 2005
Posts: 7,911
Default Re: Electron radius-quantum

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.
Reply With Quote
  #3  
Old 11-30-2007, 11:23 PM
TWCReborn TWCReborn is offline
Junior Member
 
Join Date: Nov 2007
Location: Los Angeles
Posts: 16
Default Re: Electron radius-quantum

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 <r> (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.
Reply With Quote
Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off

Forum Jump


All times are GMT -4. The time now is 10:07 PM.


Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, vBulletin Solutions Inc.