Variational Principles in Quantum Theory

Modul MA5055

Dieses Modul wird durch Fakultät für Mathematik bereitgestellt.

Diese Modulbeschreibung enthält neben den eigentlichen Beschreibungen der Inhalte, Lernergebnisse, Lehr- und Lernmethoden und Prüfungsformen auch Verweise auf die aktuellen Lehrveranstaltungen und Termine für die Modulprüfung in den jeweiligen Abschnitten.

Basisdaten

MA5055 ist ein Semestermodul in Englisch auf Master-Niveau das einmalig angeboten wird.

Die Gültigkeit des Moduls ist von SS 2016 bis WS 2016/7.

GesamtaufwandPräsenzveranstaltungenUmfang (ECTS)
150 h 45 h 5 CP

Inhalte, Lernergebnisse und Voraussetzungen

Inhalt

Quantum mechanical models were originally developed and studied in theoretical and mathematical physics, and have the mathematical form of linear partial differential equations.
More recent quantum models, such as density functional theory (DFT), nowadays play an important role in many other areas as well (chemistry, materials science, nanoscience, molecular biology). The latter model, according to a recent article on the most cited research papers of all time, is 'easily the most heavily cited concept in the physical sciences...twelve papers on the top-100 list relate to it, including 2 of the top 10' (see nature.com/top100).
The basic underlying idea is to approximate the original linear equations by NONLINEAR ones in fewer variables; this way it becomes possible to make quantitative predictions about complex systems. This is a beautiful but counter-intuitive opposite of the common strategy in undergraduate mathematics to ''linearize'' nonlinear problems.
A unifying mathematical viewpoint from which both the original and the contemporary models can be understood is a VARIATIONAL VIEWPOINT. The main task is typically to find the stationary states and energy levels of a system (atom; molecule; crystsal). Both in quantum mechanics and, say, density functional theory, this task can be formulated as minimizing a certain functional over a suitable class of functions. Taking a variational perspective allows one not just to understand basic qualitative properties of the models (e.g., existence, nonexistence, regularity, singularities of minimizers). This perspective also allows one to clarify the relationship between different models (e.g. as Gamma limits, a natural notion of convergence of variational problems), and leads in a natural way to common numerical discretization schemes.

Lernergebnisse

In this course, students will acquire a working knowledge of some modern variational methods which are transferrable to variational models in other fields. Also, students will have gained a kind of ''mathematical intuition'' for variational models of complex quantum systems, and understand basic aspects of how these models are used in the sciences.

Voraussetzungen

MA 3005 (Partial Differential Equations)
MA 3001 (Functional Analysis)
or equivalent background.
Previous familiarity with the underlying physics is NOT required.

Lehrveranstaltungen, Lern- und Lehrmethoden und Literaturhinweise

Lern- und Lehrmethoden

The module consists of 2 hours of lectures each week, supplemented by a 1 hour exercise class each week.
In the lectures, the relevant models and theoretical principles for their analysis are introduced, and illustrative examples are worked out in detail. In the exercise classes, the students analyze problems themselves which illustrate and deepen the topics of the lectures, under guidance of the class tutor.

Medienformen

Blackboard; Problem sheets

Literatur

Lecture notes uploaded after each lecture
(these are sufficient for the exam), see
http://www-m7.ma.tum.de/bin/view/Analysis/Lehre
For an interesting physics perspective (NOT followed in the course) on many of the course topics see
R. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1995

Modulprüfung

Beschreibung der Prüfungs- und Studienleistungen

The module examination is an oral exam (25 minutes). Students have to be able to state and derive basic mathematical properties of variational quantum models. They understand the variational techniques used, and are aware of how the models are utilized in some contemporary areas of the physical sciences.

Wiederholbarkeit

Eine Wiederholungsmöglichkeit wird am Semesterende angeboten.

Kondensierte Materie

Wenn Atome sich zusammen tun, wird es interessant: Grundlagenforschung an Festkörperelementen, Nanostrukturen und neuen Materialien mit überraschenden Eigenschaften treffen auf innovative Anwendungen.

Kern-, Teilchen-, Astrophysik

Ziel der Forschung ist das Verständnis unserer Welt auf subatomarem Niveau, von den Atomkernen im Zentrum der Atome bis hin zu den elementarsten Bausteinen unserer Welt.

Biophysik

Biologische Systeme, vom Protein bis hin zu lebenden Zellen und deren Verbänden, gehorchen physikalischen Prinzipien. Unser Forschungsbereich Biophysik ist deutschlandweit einer der größten Zusammenschlüsse in diesem Bereich.