Problem Set 7
Advanced Quantum Mechanics WS 13/14
Problem 1: Second Quantized OperatorsA More Formal Development
This exercise will find the second quantized form of one- and two-particle operators for
bosonic many-particle states and follows the presenta

Problem Set 13
Advanced Quantum Mechanics WS 13/14
Not a Problem: A Reminder on Covariant Notation
Let us start with a short reminderor in some cases crash courceon covariant notation.
Greek indices will go from 0 to 3 and roman indices will go from 1 to

Problem Set 11
Advanced Quantum Mechanics WS 13/14
Problem 1: Repulsive Casimir Force
In the lecture you learned, that zero point fluctuations of the electromagnetic field give an
attractive force between two parallel, perfectly conducting plates.
In this

Problem Set 10
Advanced Quantum Mechanics WS 13/14
Problem 1: Scattering Phases for the Yukawa Potential
In this problem we will consider the scattering phases for the scattering of a particle of a
Yukawa potential
a r
(1)
V (r) = V0 e a ,
r
where a is a

Problem Set 2
Advanced Quantum Mechanics WS 13/14
Problem 1: Time-Evolution of the Harmonic Oscillator
Show that the expectation values of position x and momentum p in a harmonic oscillator
follow the classical equations of motion. Would this be true if t

Problem Set 8
Advanced Quantum Mechanics WS 13/14
Bogoliubov Transformation for Fermions
In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, fermionic excitations
in a superconductor are described by a Hamiltonian of the form
H = H + H ,
(

Problem Set 12
Advanced Quantum Mechanics WS 13/14
Problem 1: Magnetic Dipole Radiation
Interactions between matter (particles), with the Hamiltonian
Hmatter =
X p2
1 X q q
+
,
2m 2 6= |r r |
and the radiation fields, with the Hamiltonian given by
X
1
Hfi

Problem Set 5
Advanced Quantum Mechanics WS 13/14
Problem 1: Second-order Fermis Golden Rule
In the lecture you discussed Fermis Golden Rule
wif =
2
|hf |V | ii|2 (Ei Ef ),
~
(1)
with the full Hamiltonian H = H0 + V and |f i and |ii being eigenstates of H

Problem Set 9
Advanced Quantum Mechanics WS 13/14
Problem 1: Hydrogen Molecule
A simple model for the Hydrogen molecule is given by the Hamiltonian
H = H0 + H1 ,
with
U
U
(n1, + n1, )2 + (n2, + n2, )2 ,
2
2
= t(c1, c2, + c2, c1, + c1, c2, + c2, c1, ),
H0

Problem Set 3
Advanced Quantum Mechanics WS 13/14
Problem 1: Coherent States of the Harmonic Oscillator
Coherent states are defined as eigenstates of the annihilation operator
a |i = |i ,
where is a complex number.
(a) Show that
2
|i = e 2 ea |0i
is a nor

Problem Set 4
Advanced Quantum Mechanics WS 13/14
Problem 1: Higher Orders of Non-Degenerate Perturbation Theory
In this exercise you will retrace the steps you saw in the lecture for the development of
non-degenerate perturbation theory.
(a) Find the exp