Variational Principles in Quantum Theory [MA5055]  WS 16/17
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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 top100 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 counterintuitive 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.
In this course, you will acquire a working knowledge of some modern variational methods which are transferrable to variational models in other fields. Also, I will try to convey a kind of ''mathematical intuition'' for variational models of complex quantum systems which does not require any previous familiarity with the underlying physics.
Prerequisites: courses in partial differential equations and functional analysis, e.g. MA 3005 and MA3001 (or equivalent)
Literature: I will upload handwritten lecture notes after each lecture. No further literature will be required.
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 2.2.2017: As agreed last week, THIS WEEK'S LECTURE WILL EXCEPTIONALLY TAKE PLACE ON THURSDAY 2.2.2017, 16:3018:00, in the usual room (Seminarraum 08.03.011).
 12.12.2016: THIS WEEK'S LECTURE WILL EXCEPTIONALLY TAKE PLACE ON MONDAY 12.12.2016, 13:1514:45, Room 03.08.022 ('Glaskasten M1/M7').
 02.08.2016: Welcome! The precise time schedule for this course will be announced shortly on this page. There will be a 2hlecture and a 1hexercise class each week.
 16.10.2016: The first lecture will be on Tuesday 18.10., 16:3018:00, 08.03.011 (Seminarraum M7/M1). Possible dates for the problem class will be discussed at the end of the lecture.
 20.10.2016: The first problem class will be on Wednesday 26. Oktober, 12:3013:15, Room 08.03.022.
Timetable
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Lectures: Tuesdays 16:3018:00, MI 08.03.011 (Seminarraum M7/M1).
Course Material
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Handwritten lecture notes will be uploaded here after each lecture.

Topic 
Lecture 1: 
Introduction; variational analysis of the hydrogen atom 
Lecture 1 
Class 1: 
Deriving EulerLagrange equations 
Sheet 1 
Lecture 2: 
Variational principle for general molecules 
Lecture 2 
Class 2: 
General variational principle; virial theorem 
Sheet 2 
Lecture 3: 
Proof that the H2 molecule binds 
Lecture 3 
Class 3: 
Galileian invariance; lower energy bound for molecules 
Sheet 3 
Lecture 4: 
Slater determinants; HartreeFock model 
Lecture 4 
Class 4: 
HartreeFock model 
Sheet 4 
Lecture 5: 
The HartreeFock equations 
Lecture 5 
Class 5: 
Density matrix formulation of HartreeFock model and HartreeFock equations 
Sheet 5 
Lecture 6: 
Banach space geometry of fermionic density matrices via creation/annihilation operators 
Lecture 6 
Class 6: 
Properties of creation/annihilation operators 
Sheet 6 
Lecture 7: 
Abstract density functional theory: Constrainedsearch definition of functional, proof of HohenbergKohn theorem, characterization of Nrepresentable densities (easy part) 
Lecture 7 
Class 7: 
Functional analytic properties of the wavefunctiontodensity map 
Sheet 7 
Lecture 8: 
Abstract density functional theory: existence of minimizers in constrainedsearch definition, characterization of Nrepresentable densities (hard part) 
Lecture 8 
Class 8: 
Existence of minimizers in ThomasFermi theory 
Sheet 8 
Lecture 9: 
Understanding the ad hoc Harriman construction via optimal transport; Scaling limits of DFT 
Lecture 9 
Lecture 10: 
Formal derivation of the KohnSham kinetic energy functional in the highdensity limit; rigorous derivation for N=2 
Lecture 10 
Class 9: 
The KohnSham equations 
Sheet 9 


Model Answer 
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