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# Variational Principles in Quantum Theory

## Module MA5055

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Basic Information

MA5055 is a semester module in English language at Master’s level which is offered once.

This module description is valid from SS 2016 to WS 2016/7.

150 h 45 h 5 CP

### Content, Learning Outcome and Preconditions

#### Content

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.

#### Learning Outcome

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.

#### Preconditions

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

### Courses, Learning and Teaching Methods and Literature

#### Learning and Teaching Methods

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.

#### Media

Blackboard; Problem sheets

#### Literature

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

### Module Exam

#### Description of exams and course work

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.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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