# 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 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.

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.

## News

• 2.2.2017: As agreed last week, THIS WEEK'S LECTURE WILL EXCEPTIONALLY TAKE PLACE ON THURSDAY 2.2.2017, 16:30-18:00, in the usual room (Seminarraum 08.03.011).

• 12.12.2016: THIS WEEK'S LECTURE WILL EXCEPTIONALLY TAKE PLACE ON MONDAY 12.12.2016, 13:15-14:45, 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 2h-lecture and a 1h-exercise class each week.

• 16.10.2016: The first lecture will be on Tuesday 18.10., 16:30-18:00, 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, 12:30-13:15, Room 08.03.022.

## Timetable

Lectures: Tuesdays 16:30-18:00, MI 08.03.011 (Seminarraum M7/M1).

## Course Material

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 Euler-Lagrange 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; Hartree-Fock model Lecture 4
Class 4: Hartree-Fock model Sheet 4
Lecture 5: The Hartree-Fock equations Lecture 5
Class 5: Density matrix formulation of Hartree-Fock model and Hartree-Fock 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: Constrained-search definition of functional, proof of Hohenberg-Kohn theorem, characterization of N-representable densities (easy part) Lecture 7
Class 7: Functional analytic properties of the wavefunction-to-density map Sheet 7
Lecture 8: Abstract density functional theory: existence of minimizers in constrained-search definition, characterization of N-representable densities (hard part) Lecture 8
Class 8: Existence of minimizers in Thomas-Fermi 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 Kohn-Sham kinetic energy functional in the high-density limit; rigorous derivation for N=2 Lecture 10
Class 9: The Kohn-Sham equations Sheet 9