Solid State Theory
Exercise 8
FS 11
Prof. M. Sigrist
Exercise 8.1
BohrvanLeeuwenTheorem
Prove the BohrvanLeeuwentheorem, which states that there is no diamagnetism in
classical physics.
Hint:
H
(
p
1
, . . . , p
N
;
q
1
, . . . , q
N
) is the Hamiltonian of the
N
particle system with vanish
ing external magnetic field. In comparison, the Hamiltonian with applied magnetic field
B
is then given by
H
(
p
1

e/cA
1
, . . . , p
N

e/cA
N
;
q
1
, . . . , q
N
), where
B
=
∇ ×
A
and
A
i
=
A
(
q
i
). The magnetization can be calculated using
M
=

∂
H
∂B
=
1
β
∂
log
Z
∂B
,
(1)
with the partition function
Z
of the system in the magnetic field.
Exercise 8.2
Landau Diamagnetism
Calculate the orbital part of the magnetization of the free electron gas in 3D in the limits
of low temperature and small external field (
T
→
0
, H
→
0). In addition, show that the
magnetic susceptibility at
T
= 0 and
H
= 0 is given by
χ
=

1
3
m
2
m
*
2
χ
P
,
(2)
where
χ
P
is the Pauli (spin)susceptibility.
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 Spring '11
 Sigrist
 Electron, Magnetism, Magnetic Field, Pauli, Landau diamagnetism, Kor K

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