Introduction to mathematical physics, fall 2013

Quantum Dynamics

Lecturer

Jani Lukkarinen

Scope

10 sp.

Type

Advanced studies

Prerequisites

Basic measure theory and analysis. Introductory courses to quantum mechanics or functional analysis are useful, but not necessary.

Lectures

Weeks 36-42 and 44-50, Monday 14-16 and Thursday 14-16 in room C124. Exercises on Mondays, 16-18 in room CK111.

Description

The main aim of the course is to present mathematical background for quantum mechanics, in particular, to explain how the time-dependent Schrödinger equation can be understood as an evolution equation on the Hilbert space of square integrable functions.

Contents: Schrödinger equation with various potentials and boundary conditions, Wigner function, unbounded operators, self-adjointness, basics of Fourier-analysis and of distribution theory, tensor product spaces, multi-particle systems and their creation operator formalism

Registration

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Links and references

Main references:

• Course lecture notes.
• M. Reed, B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis & II: Fourier Analysis, Self-Adjointness, Academic Press, 1980 & 1975.
• G. Teschl, Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, Graduate Studies in Mathematics 99, Amer. Math. Soc., 2009.
• P. L. Garrido, S. Goldstein, J. Lukkarinen, and R. Tumulka, Paradoxical Reflection in Quantum Mechanics, Am. J. Phys. 79 (2011) 1218-1231. The published version of the paper contains some misprints in the formulae in the appendices: these are correctly printed in the arxiv preprint.

Further reading:

• Wikipedia entries for the Schrödinger equation and the double slit experiment.
• Research paper about a modern double slit experiment with electrons (R. Bach, D. Pope, S.-H. Liou, and H. Batelaan, Controlled double-slit electron diffraction, New Journal of Physics 15 (2013) 033018, 7pp.) Includes movies in the supplementary data.
• Lectures notes of Roderich Tumulka on foundations and interpretation of quantum mechanics.
• Repeated (spin-)measurements: Wikipedia entry for the Stern–Gerlach experiment.
• Research paper about scaling limits of Wigner transforms (P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure App. Math. 50 (1997) 323-379)
• O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1 & 2, Springer, 2002 & 2002.
• J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, revised edition, 1994.
• S. J. Gustafson, I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Springer, 2nd edition, 2006.
• E. H. Lieb, R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2010.

Exercise session

"Final exam" = projects

As the final exam, you need to complete a project about a topic related to the course, and take part in an oral exam.  A written summary of the project should be returned to the lecturer, and when its grading has been finished, we will meet to discuss the results.  During the discussion there will be an easy oral exam. The exam consists of five questions related to the central concepts listed in the summary.

Below are some possible topics for the final exam project.  After you have chosen your topic, please contact the lecturer to get more detailed instructions.  Ideally, the report from the project should be 3-7 pages (certainly not more than 10) and contain at least one theorem with a proof based on results which are either proven in the lecture notes, or for which you can provide a reference, e.g., in a textbook.

1. Relativistic Hamiltonians (see Section 11.5.b and Exercise 12.4.)
2. Quantum particles in external magnetic fields (Section 11.5.a)
3. Rotations and spin in quantum mechanics (Section 2.20.)
4. Quadratic forms and semibounded operators (Proof of Theorem 10.5.)
5. Fock space dynamics of lattice fermions (Section 13)
6. Spectrum of the Hydrogen atom (Teschl, Chapter 10)
7. Time-dependent Hamiltonians and Dyson series
8. Trotter product formula and "Feynman path integrals" (Suggestions and instructions in Project8-v2.pdf .)

Problem sheets

The course is finished, and the solutions are no longer available on this webpage.

Lecture notes