Register to reply 
Fermidirac statistics, Griffiths 5.28 
Share this thread: 
#1
Mar1907, 08:28 PM

P: 11

1. The problem statement, all variables and given/known data
Evaluate the integrals (eqns 5.108 and 5.109) for the case of identical fermions at absolute zero. 2. Relevant equations 5.108 [tex]N=\frac{V}{2\pi^{2}}\int_{0}^{\infty}\frac{k^2}{e^{[(\hbar^{2}k^{2}/2m)\mu]/kT}+1}dk[/tex] 5.109 [tex]E=\frac{V}{2\pi^2}\frac{\hbar^2}{2m}\int_0^{\infty}\frac{k^4}{e^{[(\hbar^{2}k^{2}/2m)\mu]/kT}+1}dk[/tex] 3. The attempt at a solution Ok so at absolute zero, the chemical potential is equal to the fermi energy E_f. I'm not sure how to approach either integral because of the T dependence in the denominator in the argument of the exponential. Aren't there two cases, one for the energy of the state being above the chemical potential, and another for it being less than. If the energy is less, then the argument goes to  infinite, and the integral is just of k^2, from 0 to infinite... that doesn't seem right. If the energy is greater than mu, then the argument goes to positive infinite, and the integrand goes to 0. Fantastic. There's got to be something going on with the expressions in the argument of hte exponential to give a reasonable integrand for T=0. I think I'm missing some crucial observation. 


#3
Mar2007, 09:32 AM

P: 11

OK, so for the first integral. I have a feeling that I need to change the limits of integration because once I get to k values that exceed the FermiEnergy, the integrand goes to zero, so setting the upper limit to a specific k value will ensure convergence as well.
Oh ok, this is the probably the same method for the second integral as well. 


#4
Mar2007, 09:36 AM

Sci Advisor
HW Helper
P: 2,954

Fermidirac statistics, Griffiths 5.28



Register to reply 
Related Discussions  
The FermiDirac distribution function.  Advanced Physics Homework  5  
FermiDirac statistics valid for electron gas in metals?  Quantum Physics  5  
FermiDirac at a distance  Quantum Physics  0  
Fermi Dirac (FD) and Maxwell Boltzmann (MB)  Advanced Physics Homework  2  
FermiDirac Statistics  Introductory Physics Homework  4 