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| Semester | fall semester 2024 |
| Course frequency | Every fall sem. |
| Lecturers |
Pierre Fromholz (pierre.fromholz@unibas.ch)
Jelena Klinovaja (jelena.klinovaja@unibas.ch, Assessor) |
| Content | In physics, particularly in condensed matter, the same mathematical tools proves invaluable for tackling diverse problems. This course aims to demystify some of these tools, often mentioned briefly in other classes but seldom explored in depth. Throughout our sessions, we will learn various techniques for computing complex integrals: whether through exact methods or approximations. Additionally, we will learn to solve transcendental and differential equations exactly or employing perturbation and variational methods. In consequence, we will study various special functions in detail. Students can choose the topic of the last few classes. Past topics have included the WKB approximation, the Schrieffer-Wolf transformation, the concept of Principal value, and Wick's theorem. Each new concept and method is illustrated through at least one example drawn from the realm of physics, spanning classical Newtonian mechanics, thermodynamics, electromagnetism, wave physics, quantum mechanics, and ultimately, condensed matter. |
| Learning objectives | The content of each lecture is subject to changes depending of the pace of the class. The last lecture(s) will change depending on the wishes of the attendees (to choose the EXTRA in particular). Plan: 0. Complex integration • Residue theorem and application. • Handling branch cuts. • Laplace method for differential equations. 1. Transcendental equations • Graphical methods (determining special points). • Iteration method. • Asymptotics around special points. • Examples: magnetization of ferromagnetic (mean-field theory), energy levels of a finite (square) quantum well. 2. Integrals with small parameters • Expansion of the integrand into series. • Determining the intervals of integration giving the main contribution (and expanding the integrand there). • Estimating sums of series using integrals. 3. Method of steepest descent (for integrals and series) • Basic method and conditions. • Beta, Gamma, and Airy function; Stirling formula. • Estimating sums of series with the method. 4. Integrals with parameters and integral representations • Gauss integral and Gamma-function of half-integer argument. • Integral representation of a logarithm. • Integrals of exponential and trigonometric functions. • Integral of the Bessel function. 5. Integrals of rapidly varying functions and asymptotical expansion. • Integration of function with exponential damping. • Method of the stationary phase. • More on the Airy and the Bessel function. 6. Differential equations with small parameters • Green’s function. • Perturbation theory. • Perturbation theory in the presence of resonances. 7. Variational method • General method, notion of action. • Example: geometric optics and classical mechanics. 8. More on special functions (may be skipped on popular demand) • Dirac deltas. • Airy function. • Bessel function. • Hermite polynomial. The following lectures may vary depending of suggestions of the students and the pace of the lectures. EXTRA 1. Introduction to WKB • Introduction to the method. • Bohr-Sommerfeld quantization rule. • Double-well potential. • Quasiclassical tunneling under a barrier. EXTRA 2. The Shrieffer-Wolf transformation • Core concepts and generalization • Derivation of the t − J model EXTRA 3. More integral representations • Fourier transform of Coulomb potential. • Hubbard-Stratonovich transformation. EXTRA 4. Generalization of WKB to low-lying energy states (only if extra 1 done) • Reminder. • Splitting in double-well potential. EXTRA 5. Zoo of generalization • Multiple integrals. • Saddle-point approximation generalizations. Otherwise, these themes are also possible: Possible extensions: • Some aspects of probability theory. • Order of magnitudes estimations. • Other suggestions |
| Language of instruction | English |
| Use of digital media | No specific media used |
| Interval | Weekday | Time | Room |
|---|---|---|---|
| wöchentlich | Tuesday | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| wöchentlich | Wednesday | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Date | Time | Room |
|---|---|---|
| Tuesday 17.09.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 18.09.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 24.09.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Friday 27.09.2024 | 08.00-10.00 | Physik, Sitzungszimmer 1.09 |
| Tuesday 01.10.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 02.10.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 08.10.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 09.10.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 15.10.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 16.10.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 22.10.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 23.10.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 29.10.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 30.10.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 05.11.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 06.11.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 12.11.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 13.11.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 19.11.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 20.11.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 26.11.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 27.11.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 03.12.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 04.12.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 10.12.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 11.12.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Tuesday 17.12.2024 | 10.15-12.00 | Physik, Sitzungszimmer 1.09 |
| Wednesday 18.12.2024 | 10.15-12.00 | Physik, Seminarzimmer 4.1 |
| Modules |
Module Specialisation: Physics (Master's Studies: Nanosciences) Module: Specialisation (Master's Studies: Physics) |
| Assessment format | continuous assessment |
| Assessment registration/deregistration | Reg.: course registration, dereg: cancel course registration |
| Repeat examination | no repeat examination |
| Scale | 1-6 0,5 |
| Repeated registration | as often as necessary |
| Responsible faculty | Faculty of Science, studiendekanat-philnat@unibas.ch |
| Offered by | Departement Physik |