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Introduction to mathematical physics, spring 2017
Quantum Dynamics

 

Teacher: Jani Lukkarinen 

Scope: 10 cr

Type: Advanced studies

Teaching: Lectures: Tue 14-16 and Thu 14-16 in Exactum C123.  Exercises: see below.  "Ratkomo" tutorial time: Thu 13-14 (Exactum 3rd floor corridor)

Topics: 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

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

Course announcements

  • The course at Aalto by Ville Turunen on applications of Born-Jordan "quantization" in time-frequency analysis begins on 11.4.  More details on the course web page.

Teaching schedule

Weeks 3-9 and 11-18, Tuesday and Thursday 14-16 in Exactum, room C123.

Easter holiday 13.-19.4. 

Course material, links and references

Main references:

  • Course lecture notes (see below)
  • Brian C. Hall, Quantum Theory for Mathematicians. (Graduate Texts in Mathematics, Vol. 267) Springer, 2013

Supplementary material:

  • 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)
  • M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, 1987

  • M. de Gosson and F. Luef, Preferred quantization rules: Born–Jordan versus Weyl. The pseudo-differential point of view, J. Pseudo-Differ. Oper. Appl. 2 (2011) 115-139.  (Link to the paper, and to its arxiv-version.)
  • 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

 

Lecture notes

Registration

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Exercises

Assignments

Exercise classes

GroupDayTimeRoomInstructor
1.Fri12-14C321Brecht Donvil

"Final exam" = projects


Below are some possible topics for the final exam project (you can also suggest your own topic). Please note that in addition to the project, a minimal amount of points from the exercises or an oral exam later in May will be required to get the credits from the course.

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 13.1.)
  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 .)

Course feedback

Course feedback can be given at any point during the course. Click here.

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