PHZ7427–Solid State II
Spring 2001
Problem Set 1
Jan. 12, 2000
Due: Jan. 22, 2000
Reading: A&M Ch. 2, Mardar Chs. 6,13, Kittel Ch. 2
1. Relativistic Fermi gas. Find the Fermi energy, Fermi wave vector, and den-
sity of states for a noninteracting Fermi gas of N particles in a square 2D “box”
of side L, with spectrum
k=ck (1)
where c is a constant. Do you expect the low-temperature specific heat to differ
qualitatively from a system with parabolic spectrum, k∝k2?
2. 2nd quantized operators. Write down the 2nd-quantized form of the fol-
lowing 1st-quantized operators describing Nparticles in both a position space
basis ( ˆ
ψ(r)) and a momentum space basis (ak):
(a) particle density at r:ρ(r)=P`δ(r−r`)
(b) total number of particles: P`1=N
(c) charge current density at r:je(r)= e
2mP`[p`δ(r−r`)+δ(r−r`)p`]
(d) magnetic moment density at r:m(r)=(g/2) P`~σ`δ(r−r`)
3. Heisenberg representation. Go back to the simple harmonic oscillator Hamil-
tonian expressed in terms of raising and lowering operators:
H=¯hω(a†a+1/2) (2)
where the a’s can be thought of as Schr¨odinger operators. Show that the asso-
ciated Heisenberg operators obey canonical “2nd-quantized” commutation rela-
tions for bosons at any time, [a(t),a
†(t)] = 1. Use these facts, and the Heisen-
berg equation of motion to show a(t)=aexp(−iωt)anda†(t)=a†exp(iωt).
4. Phonon commutation relations. Given the phonon creation and annihila-
tion operators
akν=1
√N
N
X
`=1
eik·R`~kν·
sMωkν
2¯hu`+is1
2¯hMωkν
P`
(3)
a†
kν=1
√N
N
X
`=1
e−ik·R`~∗
kν·
sMωkν
2¯hu`−is1
2¯hMωkν
P`
,(4)
where the R`are the lattice site positions, the u`describe the atomic displace-
ment of the atom associated with the site R`,P`is the momentum associated
1
