### 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 (Ãœbungen):**

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 PoincarÃ© 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 (HÃ¶rerkreis):**

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

**Credits:**

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):**

English.

**Exam (PrÃ¼fung):**

There will be an oral exam (Es wird eine mÃ¼ndliche PrÃ¼fung geben).

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

**INFO ABOUT EXAM**

**Content (Inhalt):**

Section | Topic | Dates |

1. | Introduction | 23.4 |

2. | Some preliminaries and notation | " |

2.1. | Hilbert spaces | " |

2.2. | Lebesgue spaces | 26.4 |

2.3. | Fourier transform | " |

2.4. | Sobolev spaces | " |

2.5. | Forms and operators | 30.4 |

3. | A very brief mathematical formulation of classical and quantum mechanics | 3.5 |

3.1. | Some classical mechanics | 7.5 |

3.2. | The instability of classical matter | " |

3.3. | Some quantum mechanics | 10.5 |

3.4. | The one-body problem | 14.5 |

3.5. | The two-body problem and the hydrogenic atom | 17.5 |

3.6. | The N-body problem | " |

3.7. | Identical particles and quantum statistics | 21.5 |

4. | Uncertainty principles | 24.5 |

4.1. | Heisenberg | " |

4.2. | Hardy | 28.5 |

4.3. | Sobolev | 31.5 |

4.4. | Gagliardo-Nirenberg-Sobolev | " |

4.5. | Applications to stability | " |

4.6. | Poincaré | 4.6 |

4.7. | Local Sobolev-type inequalities | " |

4.8. | Local uncertainty and density formulations | 7.6 |

5. | Exclusion principles | 14.6 |

5.1. | Fermions | " |

5.2. | Weaker exclusion | " |

5.3. | Local exclusion and density formulations | " |

5.4. | Repulsive bosons | 18.6 |

5.5. | Anyons | " |

6. | The Lieb-Thirring inequality | 21.6 |

6.1. | One-body and Schrödinger formulations | " |

6.2. | Local approach to LT inequalities | 25-28.6 |

6.3. | Some direct applications of LT | 2.7 |

7. | The stability of matter | 5.7 |

7.1. | Stability of the first kind | " |

7.2. | Some electrostatics | 9-12.7 |

7.3. | Proof of stability of the second kind | 16.7 |

7.4. | Instability for bosons | 19.7 |

7.5. | Extensivity of matter | 23.7 |

- | Reserve / Extra material | 26.7 |

**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 (ErgÃ¤nzende 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 SchrÃ¶dinger 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.

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Letzte Ã„nderung: 29 July 2019

Douglas Lundholm