Department Mathematik



Vorlesung: Mathematische Quantenmechanik II (MQM2) (SoSe 2019) [16072]

Lecturer (Dozent): Prof. Dr. Douglas Lundholm

Lecture (Vorlesung):
Tue 8:30--10 & Fri 8:30--10 (in B 132).   First time (Erstes Mal): 23 April 2019.

Exercises (Übungen):
See separate webpage.

Synopsis (Kurzbeschreibung):
The aim of the course is to introduce some very important mathematical methods frequently used to solve problems in quantum mechanics, such as quantitative strong versions of the uncertainty principle of the form of Hardy, Sobolev and Poincaré inequalities, as well as general versions of the Pauli exclusion principle, leading to the celebrated Lieb-Thirring energy inequality that combines these two fundamental principles. We shall use very recent and fairly simple techniques to obtain these bounds which then are applied to give a rigorous proof of the (apparent but surprisingly subtle) stability of ordinary matter.

Audience (Hörerkreis):
Master students of Mathematics and Physics, TMP (Studierende der Mathematik, Physik, TMP).

9 (6+3) ECTS.

Prerequisites (Vorkenntnisse):
Analysis I-III, Functional Analysis (in particular Integration Theory and L^p-spaces). MQM1 or a similar course on mathematical methods in quantum mechanics is recommended, however the lecture notes will also include some basic material in mathematics and physics.

Language (Sprache):

Exam (Prüfung):
There will be an oral exam (Es wird eine mündliche Prüfung geben).

Oral exams Monday 29 July and Tuesday 30 July. Sign up via e-mail:

Content (Inhalt):
1. Introduction23.4
2. Some preliminaries and notation"
2.1.Hilbert spaces"
2.2.Lebesgue spaces26.4
2.3.Fourier transform"
2.4.Sobolev spaces"
2.5.Forms and operators30.4
3. A very brief mathematical formulation of classical and quantum mechanics3.5
3.1.Some classical mechanics7.5
3.2.The instability of classical matter"
3.3.Some quantum mechanics10.5
3.4.The one-body problem14.5
3.5.The two-body problem and the hydrogenic atom17.5
3.6.The N-body problem"
3.7.Identical particles and quantum statistics21.5
4. Uncertainty principles24.5
4.5.Applications to stability"
4.7.Local Sobolev-type inequalities"
4.8.Local uncertainty and density formulations7.6
5. Exclusion principles14.6
5.2.Weaker exclusion"
5.3.Local exclusion and density formulations"
5.4.Repulsive bosons18.6
6. The Lieb-Thirring inequality21.6
6.1.One-body and Schrödinger formulations"
6.2.Local approach to LT inequalities25-28.6
6.3.Some direct applications of LT2.7
7. The stability of matter5.7
7.1.Stability of the first kind"
7.2.Some electrostatics9-12.7
7.3.Proof of stability of the second kind16.7
7.4.Instability for bosons19.7
7.5.Extensivity of matter23.7
-Reserve / Extra material26.7

Literature (Literatur):
We will follow these lecture notes. Note that they will be revised and continuously updated during the course.

For a fairly recent and solid textbook on the topic (though lacking some of the techniques introduced in the course which are even more recent), consult:

[LS] Elliot H. Lieb and Robert Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2010. [errata etc]

Supplementary literature (Ergänzende Literatur):
  • Mikio Nakahara, Geometry, Topology and Physics, Second Edition, IOP, 2003
  • Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Volumes I - IV, 1972-2010 ca
  • Konrad Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012 [DOI]
  • Jan Philip Solovej, Many Body Quantum Mechanics, Lecture notes, 2014
  • Gerald Teschl, Mathematical Methods in Quantum Mechanics - With Applications to Schrödinger Operators, Second Edition, AMS, 2014 [PDF]
  • Walter Thirring, Classical Mathematical Physics, Third Edition, Springer, 2003
  • Walter Thirring, Quantum Mathematical Physics, Second Edition, Springer, 2003

More for the curious:
John Baez, Division Algebras and Quantum Theory
John Conway and Simon Kochen, The Strong Free Will Theorem
Jan Derezinski, Introduction to representations of the canonical commutation and anticommutation relations
Hardin and Saff, Discretizing Manifolds via Minimum Energy Points
Simon Kochen, Born's Rule, EPR, and the Free Will Theorem
Proietti et al, Experimental rejection of observer-independence in the quantum world
Carlo Rovelli, Relational Quantum Mechanics

Office hours (Sprechstunde):
By appointment via email.


Letzte Änderung: 29 July 2019

Douglas Lundholm