Fermi-dirac statistics, Griffiths 5.28

In summary, the integrals 5.108 and 5.109 can be evaluated for identical fermions at absolute zero by setting the upper limit of integration to the Fermi energy, as the integrand goes to zero for k values above the Fermi energy. This is also applicable for the energy integral, with the same method being used.
  • #1
Elvex
11
0

Homework Statement


Evaluate the integrals (eqns 5.108 and 5.109) for the case of identical fermions at absolute zero.


Homework 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]

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.
 
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  • #2
Look at equations [5.103] and [5.104].
 
  • #3
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 Fermi-Energy, 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
Elvex said:
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 Fermi-Energy, 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.

Yes, in both cases you must integrate up to [itex] k_F[/itex] only (the k value at the Fermi energy).
 

1. What is Fermi-Dirac statistics?

Fermi-Dirac statistics is a statistical model used to describe the behavior of a system of identical particles with half-integer spin, such as electrons. It takes into account the fact that these particles follow the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state simultaneously.

2. How is Fermi-Dirac statistics different from classical statistics?

In classical statistics, particles are treated as distinguishable and can occupy the same energy state simultaneously. In contrast, Fermi-Dirac statistics takes into account the indistinguishability of particles and the fact that they follow the Pauli exclusion principle.

3. What is the significance of Fermi-Dirac statistics?

Fermi-Dirac statistics is important in understanding the behavior of electrons in materials. It helps us predict the distribution of electrons in energy levels and determine the electronic properties of materials, such as conductivity and specific heat.

4. What is Griffiths 5.28 in relation to Fermi-Dirac statistics?

Griffiths 5.28 is a numerical problem in the textbook "Introduction to Quantum Mechanics" by David J. Griffiths. It involves applying Fermi-Dirac statistics to calculate the average number of particles in a system at a given temperature.

5. How is Fermi-Dirac statistics used in real-world applications?

Fermi-Dirac statistics has many practical applications, such as in the design of electronic devices like transistors and diodes. It is also used in the study of materials and their properties, as well as in the fields of astrophysics and cosmology to understand the behavior of particles in extreme conditions.

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