Quiz 2 - 02/17/03

Quiz 2 CHEM 430b/530b
Statistical Methods and Thermodynamics
02/17/03

(30 points) Exercise 1: Derive the Fermi-Dirac distribution.
(30 points) Exercise 2: Derive the Bose-Einstein distribution.
(40 points) Exercise 3: Derive an expression for the average density of electrons $\overline{N}/V$ in a metal at $T=0$ K, as a function of the Fermi energy $\mu$ and the electron mass $m$.
 
 
 
 

Solution:
Exercise 1: See topic ``Bose-Einstein and Fermi-Dirac distributions'' on pages 34 and 35 of the lecture notes.
Exercise 2: See topic ``Bose-Einstein and Fermi-Dirac distributions'' on pages 34 and 35 of the lecture notes.
Exercise 3: According to Eq. (209),

$$\overline{N} = 8 \pi \int_0^{K_f} d K K^2 \Bigg ( \frac{L}{2 \pi}......{2 \pi} \Bigg)^3 \frac{K_f^3}{3}= \frac{2 V }{(2 \pi)^3} \frac{4}{3} \pi K_f^3,$$ (282)

where $K_f$ is the Fermi momentum defined as follows

$$\frac{\hbar^2 K_f^2}{2m} =\mu.$$ (283)

Therefore,

$$\frac{\overline{N}}{V} = \frac{2}{(2 \pi)^3} \frac{4}{3} \pi \Bigg( \frac{2 m \mu}{\hbar^2} \Bigg)^{(3/2)}.$$ (284)