Statistical Mechanics - Physics at Oregon State University

4.7. PROBLEMS FOR CHAPTER 4 87
4.7 Problems for chapter 4
Problem 1.
Imagineons have weird statistics. The number of particles in a given orbital
can be 0,1, or2.
( A.) Calculate the Imagineon distribution function fI(ɛ).
( B.) Sketch the shape of this function with ɛ in units of kBT .
Fermions with spin 1
2 can also have two particles in one orbital (in the traditional
sense the word orbital does not include a spin quantum number, like the 1s
orbital in an atom).
( C.) Calculate the distribution function for the number of particles in each
orbital.
( D.) Why is this result the same/different from the result of B?
Problem 2.
Assume that for a system of N fermions the Fermi level coincides with the
energy level of M orbitals.
(A.) Calculate the entropy at T = 0.
(B.) In the thermodynamic limit N → ∞ this entropy is non-zero. What do
you know about M in this case?
Problem 3.
Starting with Ω(T, µ, V ) = −kBT
orb log Zorb(T, µ, V ) for a system of independent,
identical bosons, show that the entropy is given by
S = −kB (fBE(ɛo) log(fBE(ɛo)) − (1 + fBE(ɛo)) log(1 + fBE(ɛo))) (4.88)
Problem 4.
orb
The Maxwell distribution function fM is given by fM (ɛ; T, µ) = e 1
k B T (µ−ɛ) .
Show that