Course number: CA-MATH-806
Constructor University, Fall 2023
Much of the first part of class material follows the lecture notes by Stefan Teufel, with his kind permission:
These lecture notes have been translated to English, slightly extended, and extended with the material from the class "Advanced Topics in Quantum Mechanics" by Marcello Porta:
The standard reference for functional analysis for mathematical physicists (much of the class material is based on selected material from this book):
Other good books are:
A very good book for some of the Analysis background is:
Chapter 1: Introduction
1.1: Motivation
1.2: Single Particle QM
1.3: QM for Many Particles
Chapter 2: The Free Schrödinger Equation
2.1: Fourier Transform on S
2.2: Tempered Distributions
2.3: Long-time Asymptotics and the Momentum Operator
Chapter 3: A more general Setting for the Schrödinger Equation
3.1: Hilbert and Banach Spaces
3.2: Unitary Groups and Their Generators
3.3: Self-adjoint Operators
Chapter 4: Bose-Einstein Condensation
4.1: The Ideal Gas
4.2: Dynamics and the Nonlinear Schrödinger Equation
4.3: Fock Space and Second Quantization
4.4: Bogoliubov Theory
The grade is only based on the final exam. There is the possibility for bonus points (up to 10% in the percentage grading scheme), see below.
There will be one final exam and one make-up final exam. All topics discussed in class are relevant for the exams.
For practice, here are a midterm and a final exam from a mathematical physics class I taught in 2018. Note that there the last few classes were about unbounded operators and self-adjointness instead of mean-field dynamics:
Here you can find the final exam and a model solution from 2021.
There will be regular homework assignment. These are an integral part of the coursework and working on the exercise sheets consistently is the best preparation for the exam. As an additional motivation to work on the homework sheets there are bonus points. These are obtained in the following way: An average homework grade (in percentage scale) is computed out of all but the two worst homework sheets; dividing this by 10 gives the bonus (i.e., it can be at most 10%). Important: The bonus cannot change a failing grade into a passing grade; e.g., if you got a total bonus of 10% but only 35% in the exam, you have still failed the class with this 35% grade.
The homeworks can be handed in on the due date either in class, or they can be put into the mailbox at the entrance of Research I (on the left after you enter through the main entrance).
| Date | Sheet Number | Due Date |
|---|---|---|
| Sep 08, 2023 | Sheet 1 | Sep 20, 2023 |
| Sep 20, 2023 | Sheet 2 | Sep 27, 2023 |
| Sep 27, 2023 | Sheet 3 | Oct 04, 2023 |
| Oct 04, 2023 | Sheet 4 | Oct 11, 2023 |
| Oct 11, 2023 | Sheet 5 | Oct 18, 2023 |
| Oct 18, 2023 | Sheet 6 | Oct 25, 2023 |
| Oct 25, 2023 | Sheet 7 | Nov 01, 2023 |
| Nov 01, 2023 | Sheet 8 | Nov 09, 2023 |
| Nov 08, 2023 | Sheet 9 | Nov 16, 2023 |
| Nov 15, 2023 | Sheet 10 | Nov 23, 2023 |
| Nov 22, 2023 | Sheet 11 | Nov 30, 2023 |
| Nov 29, 2023 | Sheet 12 | Dec 07, 2023 |
Will be updated while class is progressing.
Below, please click on the date to download the lecture notes of this day.
(Note that the book references given below offer only a rough orientation. Sometimes, only parts of a particular chapter are covered in class.)
| Date | Topics |
|---|---|
| Sep. 06, 2023 | Introduction and motivation for quantum mechanics; double slit experiment See Gustafson, Sigal: Ch.s 1.1, 1.2, 1.3. |
| Sep. 08, 2023 | Introduction: Schrödinger equation for a single and many particles; brief comparison to classical mechanics See Teufel/Porta lecture notes, Ch. 1 Einleitung; see also Gustafson, Sigal Ch. 1.4. Further literature: |
| Sep. 13, 2023 | Introduction: bosons and fermions; time-independent Schrödinger equation; the Hamiltonian for non-relativistic electrons More about the derivation of the boson-fermion alternative in 3 dimensions can be found in Chapter 8.5 of the book "Bohmian Mechanics - The Physics and Mathematics of Quantum Theory" by Dürr and Teufel. More about ground states and a general brief introduction to quantum mechanics can be found in the book "The Stability of Matter in Quantum Mechanics" by Lieb and Seiringer; see Chapters 1 and 2 for the general introduction, and specifically the beginning of Chapter 2.2 for more about ground states and ground state energy. |
| Sep. 15, 2023 | L^p spaces; Lebesgue integral See Teufel/Porta lecture notes; see Lieb/Loss Ch. 2.1 for more background on Lp spaces; see Reed, Simon - I Functional Analysis, Ch. I Preliminaries for background on measure theory and the Lebesgue measure in particular. |
| Sep. 20, 2023 | Fourier transform See Teufel lecture notes. |
| Sep. 22, 2023 | Schwartz space See Teufel lecture notes; see also Reed, Simon - I Functional Analysis, parts of Chapter V.1 and the beginning of chapter V.2. |
| Sep. 27, 2023 | Schwartz space as a complete metric space; Properties of the Fourier transform on Schwartz space See Teufel lecture notes; see also Reed, Simon - I Functional Analysis, Ch. IX The Fourier Transform; see also Lieb, Loss Ch. 5 The Fourier Transform (this book does not discuss the Fourier transform on Schwartz space in detail, but aims more directly at defining it on L^p). |
| Sep. 29, 2023 | Properties of the Fourier transform on Schwartz space See references above. |
| Oct. 04, 2023 | Solution to the free Schrödinger equation in Schwartz space See Teufel lecture notes. |
| Oct. 06, 2023 | Multiplication and pseudo-differential operators and convolution See Teufel lecture notes. |
| Oct. 11, 2023 | Tempered distributions (dual spaces, examples), weak and weak* convergence, adjoints See Teufel lecture notes; see also Reed, Simon - I Functional Analysis, Ch. V.3 Functions of rapid decrease and the tempered distributions; see also Lieb, Loss, Ch. 6, where the dual space of C_c^\infty is discussed. |
| Oct. 13, 2023 | Adjoints (Fourier transform, derivatives, multiplication, and convolution for tempered distributions); Solution to the free Schrödinger equation in the sense of distributions See Teufel lecture notes. |
| Oct. 18, 2023 | Long-time asymptotics of the free Schrödinger equation and the momentum operator See Teufel lecture notes. |
| Oct. 20, 2023 | Velocity vector field; Hilbert and Banach spaces See Teufel lecture notes. For more on Hilbert and Banach spaces, see Reed, Simon - I Functional Analysis, Ch. II: Hilbert Spaces and Ch. III: Banach Spaces. See the book "Dürr, Teufel - Bohmian Mechanics. The Physics and Mathematics of Quantum Theory" for more about the particle law of motion. |
| Oct. 25, 2023 | Hilbert and Banach spaces See Teufel lecture notes. For more on Hilbert and Banach spaces, see Reed, Simon - I Functional Analysis, Ch. II: Hilbert Spaces and Ch. III: Banach Spaces. |
| Oct. 27, 2023 | Hilbert and Banach spaces; bounded operators See Teufel lecture notes; see also Reed, Simon - I Functional Analysis, Ch.s VI.1 and VI.2 (Bounded Operators). |
| Nov. 01, 2023 | Bounded operators; Fourier transform on L2; unitary operators; free propagator on L2 See Teufel lecture notes. |
| Nov. 03, 2023 | Sobolev spaces; unitary groups and their generators See Teufel lecture notes. |
| Nov. 08, 2023 | Translation operator and its generator See Teufel lecture notes. |
| Nov. 10, 2023 | Riesz Representation Theorem; Self-adjoint operators See Teufel lecture notes. |
| Nov. 15, 2023 | Bounded generators; Unbounded operators See Teufel lecture notes. For further reading: see Reed, Simon - I Functional Analysis, Ch.s VIII.1 and VIII.2 (Unbounded Operators); and Reed, Simon - II Fourier Analysis, Self-adjointness, Ch.s X.1 and X.2 (Self-adjointness and the Existence of Dynamics). |
| Nov. 17, 2023 | No class |
| Nov. 22, 2023 | Unbounded operators; Kato-Rellich; Friedrichs extension See Teufel lecture notes. |
| Nov. 24, 2023 (Session 24) | Mean-field dynamics for bosons: Hartree Theory; reduced density matrix, counting functional A good introductory paper is Pickl - A simple derivation of mean field limits for quantum systems. |
| Nov. 29, 2023 | Mean-field dynamics for bosons: reduced density matrix, counting functional See above. |
| Dec. 01, 2023 | Mean-field dynamics for bosons: Gronwall lemma See above. |
| Dec. 06, 2023 | Mean-field dynamics for bosons: Conclusion and Coulomb interaction See above. See also some previous class notes on Bogoliubov theory: Session 26, 2021, Session 27, 2021. |
| Dec. 15, 2023 | Final exam |