PHYS 6107 Spring 2009PHYS 6107 Spring 2009PHYS 6107 Spring 2009
Statistical Mechanics
Assignment #7
Due : 04/11/09 at 5pm
Problem 1, electrons in metal
Consider an ideal quantum gas of Fermi particles at a temperature T.
(a) Write the probability p(n) that there are nparticles in a given single particle state as a function of
the mean occupation number, < n >.
(b) Find the root-mean-square fluctuation <(n−< n >)2>1/2in the occupation number of a single
particle state as a function of the mean occupation number < n >. Sketch the result.
(c) g(E) is the density of states in a metal, EFis the Fermi energy. Give an expression for the total
number of electrons in the system at T= 0 in terms of EFand g(EF).
(d) Give an expression of the total number of electrons at T > 0 in terms of the chemical potential µ
and g(E), you don’t need to evaluate the integral.
Reminder: Density of state g(ǫ) is defined the same way as in classical Stat Mech. It is the number of
microstates with energy in the interval ǫand ǫ+dǫ. In other words, for quantum ideal gas, the sum
over all microstates P~
kcan be replaced by the sum over all energy spectrum. Rg(ǫ)dǫ.
Problem 2, electrons in semiconductor
(a) For a system of electrons, assumed non-interacting, show that the probability of finding an electron
in a state with energy ∆ above the chemical potential µis the same as the probability of finding an
electron absent from a state with energy ∆ below µat any given temperature T.
(b) Suppose that the density of states g(ǫ) is given by
g(ǫ) =
a(ǫ−ǫg)1/2ǫ > ǫg,
0 0 < ǫ < ǫg,
b(−ǫ)1/2ǫ < 0.
At T= 0 all states with ǫ < 0 are occupied while the other states are empty. Now for T > 0, some
states with ǫ > 0 will be occupied while some states with ǫ < 0 will be empty. If a=b, where is the
position of µ? For a6=b, write down the mathematical equation for the determination of µand discuss
qualitatively where µwill be if a > b ?a < b ?
(c) If there is an excess of ndelectrons per unit volume than can be accomodated by the states with
ǫ < 0, what is the equation for µfor T= 0? How will µshift as Tincreases?
Problem 3: Debye theory of solid
A solid made of N particles vibrating around their equilibrium positions. The Hamiltonian describing
such vibrations can be written as the sum of Hamiltonians of 3N harmonic oscilators with frequencies
ω=c~
kwhere cis the velocity of sound propagation in the solid. Each quanta of such oscillator is called
a phonon. Phonons are bosonic particles.
(a) Calculate the density of states g(ω) of the oscillators.
(b) Since there’re a total of 3N oscilators, there’s an upper cut-off frequency ωcsuch that Rωc
0g(ω)dω =
3N. Calculate this cut-off frequency. This is called Debye frequency.
(c) We can use the ideal gas of phonons to describe the solid instead of real lattice particles (ideal
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