# 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

- Lectures 17. & 19.1.
- Lecture 24.1.
- Lecture 25.1. (tensor products, new version)
- Lectures 31.1. & 2.2.
- Lectures 7. & 9.2.
- Lectures 14. & 16.2.
- Lectures 21. & 23.2. (added page 63b on 20.4.)
- Lectures 28.2., 2.3. and 14.3.
- Lecture 16.3.
- Lecture 21.3. (more details in Shubin's book or the article by de Gosson and Luef above)
- Lectures 23.3. & 28.3.
- Lectures 4.4. & 6.4. (Link to the paper about step potentials, and to a related applet about quantum tunnelling.)
- Lectures 11.4. & 20.4.
- Lectures 25.4. & 27.4. (extra material: bosons, lattice fermions)
- Summary (including info for the oral exam)

## Registration

Did you forget to register? What to do?

## Exercises

### Assignments

- Set 1, due 27.1.
- Set 2, due 3.2.
- Set 3, due 10.2.
- Set 4, due 17.2.
- Set 5, due 24.2.
- Set 6, due 3.3.
- Set 7, due
**17.3.** - Set 8, due 24.3.
- Set 9, due 31.3.
- Set 10, due 7.4.
- Set 11, due Wed 19.4.
- Set 12, due 28.4.
- Final set 13, due 5.5.

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Fri | 12-14 | C321 | Brecht 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.

- Relativistic Hamiltonians (see Section 11.5.b and Exercise 13.1.)
- Quantum particles in external magnetic fields (Section 11.5.a)
- Rotations and spin in quantum mechanics (Section 2.20.)
- Quadratic forms and semibounded operators (Proof of Theorem 10.5.)
- Fock space dynamics of lattice fermions (Section 13)
- Spectrum of the Hydrogen atom (Teschl, Chapter 10)
- Time-dependent Hamiltonians and Dyson series
- 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.