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Summer 2014

The Mathematical Structure of Quantum Mechanics

(MSc. Elective,7 ECTS points, Lecturer: David Gross, Exercises: Rafael Chaves)


There will be a lecture on Thursday, the 17h of July. Rafael will stand in on the 21st.

Course description

The success of quantum mechanics rests on the fact that it provides a calculus for accurately predicting microscopic phenomena, e.g. the wavelengths of atomic spectral lines. Introductory lectures and textbooks typically focus on this calculus. Arguably, as a consequence, the structure of the theory often remains fairly obscure. In this lecture, we take a complementary approach: We introduce a general framework for treating probabilistic theories which allows us to precisely identify differences and similarities of quantum and classical descriptions.  After laying the mathematical ground, we rigorously treat quantum mechanical phenomena like contextuality, non-locality, uncertainty relations, as well as dynamics and geometry of quantum systems.

List of topics:

  • Probabilistic theories from an operational point of view
    • preparations, transformations, measurements
    • convex geometry of state spaces, convex duality
  • Classical and quantum state spaces
  • Marginals, contextuality, non-locality
  • Uncertainty relations, information-disturbance-tradeoff, joint measureability
  • Dilation theorems, dynamics
  • Classes of quantum states
    • Entanglement
  • Brief introduction to C^*-algebras
    • Description of infinitely extended spin chains



Sheet 1 2 3 4 5 6 789 1011

Time & Venue


Mondays 2 - 4 pm weekly, SR1, physics high rise
Thursdays 12 - 2 pm bi-weekly  starting on May 8th, SR1, physics high rise




A solid understanding of linear algebra is required. A standard quantum mechanics course would be helpful, but is not strictly necessary.


A. Peres, Quantum Mechanics: Concepts and Methods
A. Holevo, Statistical Aspects of Quantum Mechanics
K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory

Lecture Notes

The notes are evolving. Current snapshot is here.

Interpretations of Quantum Mechanics (Term Paper)

Organizers: Heinz-Peter Breuer, Rafael Chaves, David Gross

Course description

Physical theories are judged on the basis of how accurate their quantitative predictions are. But beyond that, fundamental physical theories have also often provided new ways of thinking about reality. The lessons learned from quantum theory in that regard remain controversial. Many of its microscopic concepts - e.g. superposition, uncertainty relations - are not mirrored in our macroscopic perception. This has lead to a rich history of attempts to find interpretations that reconcile microscopic and macroscopic notions. Aim of this seminar is to present and assess a selection of these proposals.

Preliminary List of Topics

Literature suggestions will appear here during the course of the seminar.

  • Interpretations of Probability
  • Environment-induced Decoherence and Emergence of Classicality
  • Experimental Observations of Decoherence: From Vriuses to Gravity
  • Quantum Zeno Effect
  • Consistent History Approach to Quantum Mechanics
  • Operational Approach
    • K. Kraus, States, Effects, and Operations
  • Bell Non-Locality
  • Continuous Measurements and Monte Carlo Wave Function Method
  • Stochastic Models for the Collapse of the Wave Function
  • Bohmian mechanics
  • Many Worlds Interpretations
    • D. Deutsch, The Fabric of Reality
    • Many Worlds?, Oxford University Press
      (Individual chapters can be downloaded free of charge from the authors' websites or the arxiv.org server).
    • A critique of many world interpretations.
  • Quantum Bayesian Interpretation
  • Epistemic approaches & PBR Theorem
    • Spekken's Toy Theory?

Time & Venue

Wednesdays, 9.00 am (sharp!) Seminarraum GMH

Note: It had been considered to have an extra talk on the final day at 8.15. This is no longer an option. The final session will take place as usual.


29.04. 5.15 pm, HS1: Central assignment of students to term paper seminars.
07.05. 9.30 am Assignment of Topics
14.05. Interpretations of Probability, Dominik H├Ągele (DG)
21.05. Environment-induced Decoherence and Emergence of Classicality (HPB)
28.05. Experimental Decoherence, Valentin (HPB)
04.06. Quantum Zeno, Fred (HPB)
18.06. Bell Non-Locality, Sophie (DG)
25.06. Seminar Cancelled
02.07. Operational Approach, Evgenij (DG)
09.07. Consistent Histories, Lena (HPB)
16.07. Many Worlds, Oliver (DG)
23.07. Stochastic Collapse, Chris (HPB)
30.07. Bohmian Mechanids, Francesco (DG)

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