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Quantum Mechanics 2

Module PH1002 [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.

Module version of WS 2021/2 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2021/2WS 2020/1WS 2019/20WS 2018/9WS 2017/8WS 2016/7WS 2015/6WS 2010/1

Basic Information

PH1002 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)
300 h 90 h 10 CP

Responsible coordinator of the module PH1002 is Antonio Vairo.

Content, Learning Outcome and Preconditions


  • Time dependent Hamiltonian
    • Interaction picture
    • Two state problem: spin magnetic resonance
    • Adiabatic approximation
    • Berry's phase
  • Time dependent perturbation theory
    • Dyson series and transition probabilities
    • Constant and harmonic perturbation
    • Fermi's golden rule
    • Interaction with a classical radiation field: absorption and stimulated emission
    • Electric dipole approximation
    • Phase space and densitiy of states
    • Interaction with a classical radiation field: photoelectric effect
    • Interaction with a classical radiation field: spontaneous emission
  • Scattering theory
    • Scattering as time dependent perturbation
    • T matrix and Lippmann-Schwinger equation
    • Scattering amplitude
    • Optical theorem
    • Born approximation
    • Phase shifts and partial waves
    • Eikonal approximation
    • Low energy scattering and resonances
    • Low energy scattering and bound states
    • Scattering length, effective range, shallow bound states
    • Parity and time reversal invariance
    • Inelastic electron-atom scattering; form factors
  • Open quantum systems
    • Entangled states
    • Density matrix
    • Reduced density matrix
    • von Neumann entropy
    • Unitary evolution: von Neumann equation
    • Evolution of the reduced density matrix: Lindblad equation
    • Quantum cryptography and no cloning theorem
  • Systems of identical particles
    • Bosons and fermions
    • Two-electron system
    • Helium atom
    • Scattering of identical particles
    • Multiparticle states
  • Fock space and second quantization
    • Fock space for bosons
    • Fock space for fermions
    • One particle and multiparticle operators
    • Field operators
    • Equation of motion
    • Fermi energy and electron gas
  • Quantization of the electromagnetic field
    • Photons
    • Casimir effect
  • 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
    • Symmetries: Lorentz, C, P, T
    • Solution of the Dirac equation in a central potential; energy spectrum of the hydrogen atom

Learning Outcome

After successfully taking part in this module the students are able to:

1- Derive Fermi's golden rule and apply it to calculate transition probabilities,

2- Calculate the scattering amplitude and the differential cross-section in a scattering process,

3- Derive the optical theorem and understand its consequences,

4- Define entagled states. Define mixtures and pure states. Write the evolution equations for closed and open quantum systems,

5- Write-down the wave function of a system of bosons or fermions,

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

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

8- Understand the formalism of "second quantization",

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


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

VO 4 Quantum Mechanics 2 Vairo, A. Wed, 12:00–14:00, virtuell
Fri, 10:00–12:00, virtuell
UE 2 Exercise to Quantum Mechanics 2 Qerimi, G.
Responsible/Coordination: Vairo, A.
dates in groups eLearning

Learning and Teaching Methods

The modul consists of a lecture and exercise classes.

In the thematically structured lecture the learning content is presented. With cross references between different topics the universal concepts in physics are shown. In scientific discussions the students are involved to stimulate their analytic-physics intellectual power.

In the exercise the learning content is deepened and exercised using problem examples and calculations. Thus the students are able to explain and apply the learned physics knowledge independently.


Blackboard/online (Zoom), Script, 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:

  • Calculate transition probabilities for a harmonic oscillator under a small time-dependent perturbation.
  • Derive the selection rules and transition rates for a hydrogen atom under the influence of a radiation field.
  • Calculate the phase shifts and the cross section for a non-relativistic particle scattering off a given potential.

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.

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