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Quantum Mechanics 2

Module NAT3001 [ThPh KTA]

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

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 Elite-Master Program Theoretical and Mathematical Physics (TMP)

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workloadContact hoursCredits (ECTS)
150 h 90 h 5 CP

Responsible coordinator of the module NAT3001 is Nora Brambilla.

Content, Learning Outcome and Preconditions


  • Path integral
    • classical and semiclassical limit
    • Aharonov-Bohm 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
    • two-level 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: Klein-Gordon and Dirac equations
    • Klein-Gordon 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: spin-orbit interaction and Darwin term for the hydrogen atom, spectrumMolecular
    • Lamb shift effect
  • Molecular states: Born-Oppenheimer 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- Write-down the wave function of a system of bosons or fermions,

3- Write-down the Klein-Gordon and Dirac equations,

4- Calculate the Dirac energy spectrum for the hydrogen atom,

5- Understand the formalism of "second quantization",

6- Quantize the electromagnetic, Klein-Gordon and Dirac field operators

7- Use the number occupation representation and Fock space

8- Understand decoherence


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

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.


Blackboard/online (Zoom), slides if available


    C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill 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. Cohen-Tannoudji, 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 mid-term 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.

Current exam dates

Currently TUMonline lists the following exam dates. In addition to the general information above please refer to the current information given during the course.

Exam to Quantum Mechanics 2
Thu, 2023-02-16, 14:15 till 16:15 102
till 2023-01-15 (cancelation of registration till 2023-02-09)
Tue, 2023-04-04, 11:00 till 13:00 1350
till 2023-01-15 (cancelation of registration till 2023-03-28)
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