Phys 132 Homework 20 Solutions
Professor Crystal Martin
TA: Ellie Hadjiyska
Problem 20: (Problem 2.1) We are asked to consider an ideal gas of degenerate, non-
relativistic electrons with concentration n.
(a) We first want to find an expression for the Fermi energy F(the maximum energy of any electron
in gas). We can find the Fermi momentum pF(maximum momentum of any electron in gas) by
summing over all momentum states within the gas up to this maximum momentum pF(Phillips
Eq. (2.26)):
N=ZpF
0
gs
V
h34πp2dp =8πV
3h3p3
F
where gs= 2 for the two independent spin states of an electron. This implies that the Fermi
momentum is (Phillips Eq. (2.27))
pF=3n
8π1/3
h
The Fermi energy is therefore
F=p2
F
2me
=h2
2me3n
8π2/3
=~2
2me
(3π2n)2/3
where ~=h/(2π). We now assume that there is a temperature Tassociated with the gas and the
quantum concentration nQ(Phillips Eq. (2.22)) equals the actual concentration nor
n=nQ=2πmekT
h23/2
Combining these expressions gives
F=~2
2me"3π22πmekT
h23/2#2/3
=(3π2)2/3
4πkT =9π
64 1/3
kT
So the ratio of kT to the Fermi energy is
kT
F
=64
9π1/3
≈1.313
