Jacobs University, Spring 2020
Much of the 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:
Other good books are:
A very good book for some of the Analysis background is:
THE standard reference for functional analysis for mathematical physicists:
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: The Schrödinger Equation with Potential
3.1: Hilbert and Banach Spaces
3.2: Unitary Groups and Their Generators
3.3: Self-adjoint Operators
The grade is only based on the final exam.
There will be one final exam and one make-up final exam. All topics discussed in class are relevant for the exams. Note that this class uses gradescope for grading exams, see here for more information. (Note: Just talk to the instructor if you prefer that gradescope is not used for your exam.) The final exam will be a take-home exam. It will be given out on May 18, 2020, and has to be handed in on or before May 27, 2020, 23:59.
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 exams. Please go to Jacobs moodle to view a sketch of the solutions.
| Date | Sheet Number | Due Date |
|---|---|---|
| Feb 10, 2020 | Sheet 1 | Feb 17, 2020 |
| Feb 17, 2020 | Sheet 2 | Feb 24, 2020 |
| Feb 24, 2020 | Sheet 3 | Mar 02, 2020 |
| Mar 09, 2020 | Sheet 4 | Mar 16, 2020 |
| Mar 16, 2020 | Sheet 5 | Mar 23, 2020 |
| Mar 23, 2020 | Sheet 6 | Mar 30, 2020 |
| Apr 06, 2020 | Sheet 7 | Apr 20, 2020 |
| Apr 20, 2020 | Sheet 8 | Apr 27, 2020 |
| Apr 27, 2020 | Sheet 9 | May 04, 2020 |
| May 04, 2020 | Sheet 10 | May 11, 2020 |
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 |
|---|---|
| Feb. 03, 2020 | Introduction and motivation for quantum mechanics (double slit); Single particle QM See Gustafson, Sigal: Ch.s 1.1, 1.2, 1.3. |
| Feb. 05, 2020 | Introduction: Schrödinger equation for a single and many particles; brief comparison to classical mechanics See Teufel lecture notes, Ch. 1 Einleitung; see also Gustafson, Sigal Ch. 1.4. Further literature: |
| Feb. 10, 2020 | 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. |
| Feb. 12, 2020 | The Free Schrödinger Equation; Fourier transform See Teufel 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. |
| Feb. 17, 2020 | Schwartz space as a complete metric space See Teufel lecture notes. |
| Feb. 19, 2020 | Properties of the Fourier transform on the Schwartz space; solution to the free Schrödinger equation in the 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 the Schwartz space in detail, but aims more directly at defining it on L^p). |
| Feb. 24, 2020 | Pseudo-differential operators See Teufel lecture notes. |
| Feb. 26, 2020 | Tempered distributions (dual spaces, examples, weak and weak* convergence) 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. |
| Mar. 02, 2020 | Adjoints (Fourier transform, derivatives, multiplication, and convolution for tempered distributions) See Teufel lecture notes. |
| Mar. 04, 2020 | Long-time asymptotics of the free Schrödinger equation and the momentum operator See Teufel lecture notes. |
| Mar. 09, 2020 | Long-time asymptotics of the free Schrödinger equation and the momentum operator See Teufel lecture notes. |
| Mar. 11, 2020 | 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. |
| Mar. 16, 2020 | Hilbert and Banach spaces See Teufel lecture notes. |
| Mar. 18, 2020 | Bounded operators See Teufel lecture notes; see also Reed, Simon - I Functional Analysis, Ch.s VI.1 and VI.2 (Bounded Operators). |
| Mar. 23, 2020 | Bounded operators and convergence See Teufel lecture notes. |
| Mar. 25, 2020 | Unitary groups and their generators; Sobolev spaces See Teufel lecture notes. |
| Mar. 30, 2020 | Translation operator and its generator; Riesz Representation Theorem See Teufel lecture notes. |
| Apr. 01, 2020 | Self-adjoint operators See Teufel lecture notes. |
| Apr. 06, 2020 | no class (Spring Break) |
| Apr. 08, 2020 | no class (Spring Break) |
| Apr. 13, 2020 | no class (Spring Break) |
| Apr. 15, 2020 | Self-adjoint operators; unbounded operators See Teufel lecture notes; see also Reed, Simon - I Functional Analysis, Ch.s VIII.1 and VIII.2 (Unbounded Operators). |
| Apr. 20, 2020 | Closed unbounded operators See Teufel lecture notes; see also these notes for more on closable/closed operators. |
| Apr. 22, 2020 | Essential self-adjointness See Teufel lecture notes. |
| Apr. 27, 2020 | Criteria for (essential) self-adjointness See Teufel lecture notes; see also Reed, Simon - II Fourier Analysis, Self-adjointness, Ch.s X.1 and X.2 (Self-adjointness and the Existence of Dynamics). |
| Apr. 29, 2020 | Cayley transform and self-adjointness See Teufel lecture notes. |
| May 04, 2020 | Resolvent and self-adjointness See Teufel lecture notes. |
| May 06, 2020 | Kato-Rellich See Teufel lecture notes. |
| May 11, 2020 | Exercises and review |
| May 13, 2020 | Exercises and review |
| May 18-27, 2020 | Take-home Final Exam |