Prof. Dr. Roser Valent´ı
M. Altmeyer, S. Backes, K. Riedl
Exercises for Advanced Solid State Theory
Sommersemester 2015
Problem set 5,
Due date: Wednesday, 20.05.15
Exercise 1: Landau eigenstates and eigenenergies (3 points)
Find the solution to the Hamiltonian for a free electron diamagnetically coupled to an
external magnetic field B,
2
2 2
2
pˆ2
ˆ = pˆx + y + pˆz − eB pˆy xˆ + e B xˆ2 ,
H
2m 2m 2m mc
2mc2
and show that the eigenstates and eigenenergies of H are given by, respectively,
~2 kz2
1
En,kz = ~ω0 n +
+
2
2m
and
Ψn,ky ,kz (~r) = cHn
x − x0
λ
e−
(x−x0 )2
2λ2
eiky y eikz z ,
(1)
(2)
(3)
where
r
eB
~ky
~
ω0 =
, x0 =
, λ=
,
(4)
mc
mω0
mω0
c is a constant and Hn are the Hermite polynomials.
ˆ commutes with both pˆy and pˆz and then use a proper product
(Hint: Show first that H
ansatz for the wavefunction.)
Exercise 2: Hall resistivity for the Landau levels (7 points)
(a) Show that the expectation value of the current density operator
1 e Jy = ney˙ = ne
py − Ay , n = electron concentration,
m
c
(5)
over the first Landau level
(x−x0 )2
1
eiky y eikz z
e− 2λ2
Ψ0,kx ,kz (~r) = p √ √
Ly Lz π 1/2 λ
is zero.
1
(6)
(b) Add to the Hamiltonian for the Landau magnetism
2
p2z
p2x
m 2
py
H=
+
+ ω0 x −
2m 2m
2
mω0
(7)
a term corresponding to an electrical field along the x-direction,
HE = −eEx,
(8)
and solve the new Hamiltonian H + HE . In particular, sketch the Landau levels as
a function of ky .
(Hint: You can rewrite the Hamiltonian as the one of a harmonic oscillator centered
around a new position x1 .)
(c) Calculate now the expectation value of the current density operator over the shifted
harmonic oscillator that you obtained as the ground state of H + HE , showing that
you recover the classical result:
hJy i =
nec
E.
B
(9)
(d) Show that the classical result for the Hall resistance equals the quantum case if the
electron density n corresponds to completely filled Landau levels.
[Hint: (i) Consider electron gas with n = Lx LNy Lz , that creates the Hall voltage
IB
UH = necL
and (ii) assume that N = dν, where d is the degeneracy of the Landau
z
levels and ν is the Landau level filling.]
2