Average Kinetic Energy of Electron in the Conduction Band

  • #1
Teymur
16
3
Homework Statement
Show that:
Relevant Equations


Hello,
I've seen in a few books on solid state physics that one can deduce an expression for average K.E.:



from the following:



I can't, however, find any work through of how to do so. I've had a go at the bottom part:

where and

and

and

to get:



But how does one integrate the numerator with the term and simplify to the desired result?
 
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  • #2
p.s. I used: for the integral:
 
  • #3
Teymur said:
But how does one integrate the numerator with the term and simplify to the desired result?
The numerator integration is very similar to the integration in the denominator. The factor has a simple relation to .
 
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  • #4
Aha .. I'm not sure why I didn't spot that.
 
  • #5
Another very important trick used in statistical physics is to calculate the denominator, the socalled "partition sum" and then take a derivative wrt. , which is an application of the celebrated Feynman-Hellmann theorem.
 
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