Complex quantum systems : analysis of large coulomb systems / editor, Heinz Siedentop, Ludwig-Maximilians-Universitat, Munchen, Germany.

Format:

Book

Contributor:

Siedentop, Heinz Karl Heinrich, editor.

Series:

Lecture notes series (National University of Singapore. Institute for Mathematical Sciences) ; v. 24.

Lecture notes series, 1793-0758 ; v. 24

Language:

English

Subjects (All):

Quantum electrodynamics--Mathematics.

Quantum electrodynamics.

Quantum theory--Congresses.

Quantum theory.

Physical Description:

1 online resource (xi, 290 pages) : illustrations.

Edition:

1st ed.

Other Title:

Quantum systems

Coulomb systems

Place of Publication:

New Jersey : World Scientific, [2013]

Language Note:

English

Summary:

This volume is based on lectures given during the program Complex Quantum Systems held at the National University of Singapore's Institute for Mathematical Sciences from 17 February to 27 March 2010. It guides the reader through two introductory expositions on large Coulomb systems to five of the most important developments in the field: derivation of mean field equations, derivation of effective Hamiltonians, alternative high precision methods in quantum chemistry, modern many body methods originating from quantum information, and - the most complex - semirelativistic quantum electrodynamics.

Contents:

Intro

CONTENTS

Foreword

Preface

Stability of Matter Rafael D. Benguria and Benjamın A. Loewe

1. Introduction: The stability of quantum systems: A historical overview

2. Stability of Matter: The classical proof of Lieb and Thirring

2.1. Stability of the hydrogen atom in non-relativistic quantum mechanics

2.2. Stability of a system of N electrons in non-relativistic quantum mechanics

2.3. Stability of a many particle system via Thomas-Fermi theory

2.4. Bibliographical remarks

3. Lieb-Thirring Inequalities

3.1. Use of commutation methods to prove the Lieb-Thirring inequality for = 3/2 in dimension 1

3.2. The Eden-Foias bound ([46])

3.3. Bibliographical remarks

4. Electrostatic Inequalities

5. The Maximum Number of Electrons an Atom Can Bind

5.1. The maximum number of electrons for a one center case in the Thomas-Fermi model

5.2. Bound on Nc(Z) for the TFW model in the atomic case

6. The Stability of Matter for a Relativistic Toy Model

6.1. Bibliographical remarks

7. A New Lieb-Oxford Bound with Gradient Corrections

Acknowledgments

Appendix: A Short History of the Atom

References

Mathematical Density and Density Matrix Functional Theory (DFT and DMFT) Volker Bach

1. Introduction

2. Exchange Correlation and LDA

3. Kinetic Energy and Lieb-Thirring Inequality

4. Thomas-Fermi Theory and Stability of Matter

5. Hartree-Fock Theory

6. Correlation Estimate Improving the Lieb-Oxford Inequality

7. Accuracy of the Hartree-Fock Approximation for Large Neutral Atoms

8. N-Representability

On the Dynamics of a Fermi Gas in a Random Medium with Dynamical Hartree-Fock Interactions Thomas Chen

Acknowledgment

2. Fermi Gas in a Random Medium

2.1. Statement of the main results.

2.2. Boltzmann limit of the momentum distribution function

2.3. Outline of the proof

2.4. Feynman graph expansion

2.5. Classification of graphs

2.6. Discussion of the result

3. Persistence of Quasifreeness in the Boltzmann Limit

3.1. Outline of the proof of Theorem 3.1

3.1.1. Completely disconnected graphs

3.1.2. Non-disconnected graphs

4. Fermi Gas with Dynamical Hartree-Fock Interactions

4.1. Statement of main results

4.1.1. The regime λ ≤ Cη2

4.1.2. The regime η = o(√λ)

4.1.3. The regime t = T/η2 and λ = Oη(1)

On the Minimization of Hamiltonians over Pure Gaussian States Jan Derezinski, Marcin Napiorkowski, and Jan Philip Solovej

2. Preliminaries

2.1. 2nd quantization

2.2. Wick quantization

2.3. Bogoliubov transformations

2.4. Pure Gaussian states

3. Main Result

Variational Approach to Electronic Structure Calculations on Second-Order Reduced Density Matrices and the N-Representability Problem Maho Nakata, Mituhiro Fukuda, and Katsuki Fujisawa

2. The Reduced-Density-Matrix Method

2.1. Pure states and ensemble states

2.2. The first-order and second-order reduced density matrices

2.2.1. Coordinate representation

2.2.2. Second-quantized representation

2.2.3. Equivalence between the coordinate and second-quantized representations

2.2.4. Some properties of 1- and 2-RDMs

2.3. Solving the ground state problem using 1- and 2-RDMs

2.4. The N-representability problem and the N-representability conditions

2.5. On the complete N-representability conditions

2.6. Formulating the variational problem and its geometrical representation

2.7. Some of the known necessary N-representability conditions

2.8. The reduced-density-matrix method

2.9. Interpreting the conditions.

3. Formulating the RDM Problem as a Semidefinite Program and its Solution Using the Interior-Point Method

3.1. Semidefinite program

3.2. Formulation of the RDM problem as an SDP

3.3. Theoretical computational complexity of the primal-dual interior-point method

4. Some Historical Remarks

5. Numerical Results for the RDM Method

5.1. New numerical results for larger systems

5.2. Summary of the numerical experiments

6. Concluding Remarks

Fermionic Quantum Many-Body Systems: A Quantum Information Approach Christina V. Kraus

2. Pairing in Fermionic Systems: A Quantum Information Perspective

2.1. Motivation

2.2. Pairing theory

2.3. Detection and quantification of pairing

2.3.1. Detection of pairing

2.4. Examples: Fermionic Gaussian states and number-conserving states

2.4.1. Pairing of Gaussian states

2.4.2. Pairing of number-conserving states

2.5. Pairing as a resource

3. Fermionic Projected Entangled Pair State

3.1. A review of the PEPS-construction

3.2. Construction of fPEPS

3.3. Relation between fPEPS and PEPS

3.4. Examples

4. Conclusion and Outlook

Hydrogen-Like Atoms in Relativistic QED Martin Konenberg, Oliver Matte, and Edgardo Stockmeyer

2. Definition of the Models

2.1. Operators in Fock-space

2.2. Interaction term

2.3. The semi-relativistic Pauli-Fierz and no-pair Hamiltonians

2.4. How to deal with the non-local terms

3. Self-Adjointness

3.1. Diamagnetic inequalities in QED

3.2. Semi-boundedness

4. Bounds on the Ionization Energy

5. Exponential Localization

5.1. A general strategy to prove the localization of spectral subspaces

5.2. Choice of the comparison operator Y

5.3. Conjugation of Y with exponential weights.

6. Existence of Ground States with Mass

6.1. Operators with photon mass

6.2. Discretization of the photon momenta

6.3. Comparison of operators with different coupling functions

6.4. Higher order estimates and their consequences

6.5. Continuity of the ionization thresholds and ground state energies

6.6. Proofs of the existence of ground states with mass

7. Infra-Red Bounds

7.1. The gauge transformed operator

7.2. Soft photon bound for the semi-relativistic Pauli-Fierz operator

8. Existence of Ground States

8.1. Ground states without photon mass

8.2. Ground state degeneracy

9. Commutator Estimates

9.1. Basic estimates

9.2. Commuting projections with the field energy

9.3. Double commutators

References.

Notes:

Description based upon print version of record.

Includes bibliographical references.

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Siedentop, Heinz Complex Quantum Systems: Analysis Of Large Coulomb Systems

ISBN:

9789814460156

981446015X

OCLC:

853679950