Quantum Mechanics 2
Module NAT3001 [ThPh KTA]
Basic Information
NAT3001 is a semester module in English or German language at Master’s level which is offered in winter semester.
This Module is included in the following catalogues within the study programs in physics.
 Theory courses for nuclear, particle, and astrophysics
 Complementary catalogue of special courses for condensed matter physics
 Complementary catalogue of special courses for Biophysics
 Complementary catalogue of special courses for Applied and Engineering Physics
 Specialization Modules in EliteMaster Program Theoretical and Mathematical Physics (TMP)
If not stated otherwise for export to a nonphysics program the student workload is given in the following table.
Total workload  Contact hours  Credits (ECTS) 

150 h  90 h  5 CP 
Responsible coordinator of the module NAT3001 is Nora Brambilla.
Content, Learning Outcome and Preconditions
Content
 Path integral
 classical and semiclassical limit
 AharonovBohm effect
 Open quantum systems
 density matrix, unitary evolution and von Neumann equation
 reduced density matrix
 evolution of the reduced density matrix and Linblad equation
 decoherence
 twolevel system in Linblad formalism
 Systems of identical particles: bosons and fermions
 Fock space and second quantization
 one particle and multiparticle operators
 Field operators
 Equation of motion
 Fermi energy and electron gas
 One particle relativistic equations: KleinGordon and Dirac equations
 KleinGordon equation: derivation and solutions
 Hamilton, momentum, charge current operators in second quantization
 Dirac equation
 Derivation
 Dirac matrices
 Free particle solution
 Coupling to electromagnetism, Pauli equation, gyromagnetic ratio
 Quantized field
 Solution of the Dirac equation in a central potential
 Nonrelativistc expansion of the relativistic equations: spinorbit interaction and Darwin term for the hydrogen atom, spectrumMolecular
 Lamb shift effect
 Molecular states: BornOppenheimer approximation and van Der Waals forces (if time permits)
Learning Outcome
After successfully taking part in this module the students are able to:
1 Define entagled states. Define mixtures and pure states. Write the evolution equations for closed and open quantum systems,
2 Writedown the wave function of a system of bosons or fermions,
3 Writedown the KleinGordon and Dirac equations,
4 Calculate the Dirac energy spectrum for the hydrogen atom,
5 Understand the formalism of "second quantization",
6 Quantize the electromagnetic, KleinGordon and Dirac field operators
7 Use the number occupation representation and Fock space
8 Understand decoherence
Preconditions
No prerequisites that are not already included in the prerequisites for the Master’s programmes.
Courses, Learning and Teaching Methods and Literature
Courses and Schedule
Type  SWS  Title  Lecturer(s)  Dates  Links 

VO  2  Quantum Mechanics 2  Brambilla, N. 
Fri, 10:00–12:00, PH HS2 
eLearning 
UE  2  Exercise to Quantum Mechanics 2 
Mohapatra, A.
Säppi, S.
Responsible/Coordination: Brambilla, N. 
dates in groups 
eLearning 
Learning and Teaching Methods
In the lecture the subjects will be explained in details, the theory will be constructed and examples and applications will be worked out. Discussion with students will be encouraged. The students will be guided through the literature.
In the exercises the students will be in small groups and the content of the exercises will be discussed together. In the exercise the learning content is deepened and understood using problem examples and calculations. Thus the students will be able to explain and apply the learned physics knowledge independently.
Media
Blackboard/online (Zoom), slides if available
Literature

C. Itzykson and J.B. Zuber,
Quantum Field Theory,
McGrawHill 1980
R.P. Feynman, Feynman Vorlesungen über Physik Bd. 3, Oldenbourg Wissenschaftsverlag 1999
S. Flügge, Practical Quantum Mechanics, Springer 1999
A. Messiah, Quantenmechanik Bd. 1 und 2, de Gruyter 1991
A. Galindo and P. Pascual, Quantum Mechanics I und II, Springer 1990
F. Schwabl, Quantenmechanik, Springer 2007
F. Schwabl, Quantenmechanik für Fortgeschrittene, Springer 2008
G. Auletta, M. Fortunato and G. Parisi, Quantum Mechanics, Cambridge University Press 2009
M. Le Bellac, Quantum Physics, Cambridge University Press 2011
S. Weinberg, Lectures on Quantum Mechanics, Cambridge University Press 2015
C. CohenTannoudji, B. Diu and Franck Laloë Quantenmechanik Bd. 1, 2 und 3, de Gruyter 2019
J.J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Cambridge University Press 2020
Module Exam
Description of exams and course work
There will be a written exam of 90 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.
For example an assignment in the exam might be:
 Derive the Linblad equation
 Use the open quantum system formalism to solve a two state system problem
 Calculate the classical limit of a path integral formulation
 Solve the Dirac equation in a central potential
 Calculate the relativistic corrections to the hydrogen spectrum
 Calculate properties of a system of identical particles
 Use the Fock space and second quantization to calculate properties of a system of bosons or fermions
In the exam no learning aids are permitted.
Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.
There will be a bonus (one intermediate stepping of "0,3" to the better grade) on passed module exams (4,3 is not upgraded to 4,0). The bonus is applicable to both exam period directly following the lecture period and subject to the condition that the student passes the midterm of of obtaining 60% of the total points of the problem sets.
Exam Repetition
The exam may be repeated at the end of the semester.