Quantum Mechanics with Applications to Nanotechnology and Information Science

Quantum mechanics transcends and supplants classical mechanics at the atomic and subatomic levels. It provides the underlying framework for many subfields of physics, chemistry and materials science, including condensed matter physics, atomic physics, molecular physics, quantum chemistry, particle physics, and nuclear physics. It is the only way we can understand the structure of materials, from the semiconductors in our computers to the metal in our automobiles. It is also the scaffolding supporting much of nanoscience and nanotechnology. The purpose of this book is to present the fundamentals of quantum theory within a modern perspective, with emphasis on applications to nanoscience and nanotechnology, and information-technology. As the frontiers of science have advanced, the sort of curriculum adequate for students in the sciences and engineering twenty years ago is no longer satisfactory today. Hence, the emphasis on new topics that are not included in older reference texts, such as quantum information theory, decoherence and dissipation, and on applications to nanotechnology, including quantum dots, wires and wells.

• This book provides a novel approach to Quantum Mechanics whilst also giving readers the requisite background and training for the scientists and engineers of the 21st Century who need to come to grips with quantum phenomena
• The fundamentals of quantum theory are provided within a modern perspective, with emphasis on applications to nanoscience and nanotechnology, and information-technology
• Older books on quantum mechanics do not contain the amalgam of ideas, concepts and tools necessary to prepare engineers and scientists to deal with the new facets of quantum mechanics and their application to quantum information science and nanotechnology
• As the frontiers of science have advanced, the sort of curriculum adequate for students in the sciences and engineering twenty years ago is no longer satisfactory today
• There are many excellent quantum mechanics books available, but none have the emphasis on nanotechnology and quantum information science that this book has

Teaching and research faculty, upper-undergraduate and graduate students majoring in Physics, Chemistry, Chemical Engineering, Material Engineering, Electrical Engineering

Preface

Acknowledgments

1. Introduction to Quantum Mechanics

1.1 What is Quantum Mechanics?

1.2 Nanotechnology and Information Technology

1.3 A First Taste of Quantum Mechanics

2. The Formalism of Quantum Mechanics

2.1 Hilbert Space and Dirac Notation

2.2 Hermitian and Anti-Hermitian Operators

2.3 The Uncertainty Principle

2.4 The Measurement Problem

2.5 Mixed States: Density Matrix Formulation

2.6 The Wigner Representation

2.7 Schrödinger and Heisenberg Representations

2.8 The Correspondence Principle and the Classical Limit

2.9 Symmetry and Conservation Laws in Quantum Mechanics

3. Angular Momentum and Spherical Symmetry

3.1 Angular Momentum in Quantum Mechanics

3.2 Spherically Symmetric Systems

3.3 Rotations and Angular Momentum

3.4 Addition (Coupling) of Angular Momenta

3.5 Tensor Operators

3.6 Symmetry Considerations

4. Spin

4.1 Spin Angular Momentum

4.2 Spinors

4.3 Electron in a Magnetic Field

4.4 Time-Reversal Properties of Spinors

4.5 Spin–Orbit Interaction in Atoms

4.6 Hyperfine Interaction

4.7 Spin-Dipolar Interactions

4.8 Introduction to Magnetic Resonance

5. Quantum Information

5.1 Classical Computation and Classical Information

5.2 Quantum Information

5.3 Quantum Computing Algorithms

5.4 Decoherence

5.5 Quantum Error Correction

5.6 Experimental Implementations

5.7 The EPR Paradox

5.8 Bell’s Inequalities

6. Quantum Dynamics and Correlations

6.1 Two-Level Systems

6.2 Three-Level Systems

6.3 Classification of Correlation and Entanglement

6.4 Three-Level System Dynamics

6.5 Continuous-Variable Systems

6.6 Wave Packet Dynamics

6.7 Time-Dependent Hamiltonians

6.8 Quantum Optimal Control Theory

7. Approximation Methods

7.1 Basis-State Expansions

7.2 Semiclassical Approximations

7.3 Perturbation Theory

7.4 Dynamics in an Electromagnetic Field

7.5 Exponential and Nonexponential Decay

7.6 The Variational Method

7.7 The Sudden Approximation

7.8 The Adiabatic Approximation

7.9 Linear Response Theory

8. Identical Particles

8.1 Permutation Symmetry

8.2 Exchange Symmetry

8.3 Permanents and Slater Determinants

8.4 Simple Two- and Three-Electron States

8.5 Exchange Symmetry for Two Two-Level Systems

8.6 Many-Particle Exchange Symmetry

9. Electronic Properties of Solids

9.1 The Free Electron Gas

9.2 Elementary Theories of Conductivity

9.3 Crystal Structure

9.4 Electrons in a Periodic Potential

9.5 Magnetic Field Effects

9.6 Semiconductors

9.7 Spintronics

9.8 Low-Energy Excitations

9.9 Insulators

10. Electronic Structure of Multielectron Systems

10.1 The Multielectron System Hamiltonian

10.2 Slater and Gaussian Type Atomic Orbitals

10.3 Term Symbols for Atoms

10.4 Two-Electron Systems

10.5 Hartree Approximation for Multielectron Systems

10.6 The Hartree–Fock Method

10.7 Koopmans’ Theorem

10.8 Atomic Radii

10.9 Multielectron Fine Structure: Hund’s Rules

10.10 Electronic Structure of Molecules

10.11 Hartree–Fock for Metals

10.12 Electron Correlation

11. Molecules

11.1 Molecular Symmetries

11.2 Diatomic Electronic States

11.3 The Born-Oppenheimer Approximation

11.4 Rotational and Vibrational Structure

11.5 Vibrational Modes and Symmetry

11.6 Selection Rules for Optical Transitions

11.7 The Franck–Condon Principle

12. Scattering Theory

12.1 Classical Scattering Theory

12.2 Quantum Scattering

12.3 Stationary Scattering Theory

12.4 Aspects of Formal Scattering Theory

12.5 Central Potentials

12.6 Resonance Scattering

12.7 Approximation Methods

12.8 Particles with Internal Degrees of Freedom

12.9 Scattering in Low-Dimensional Systems

13. Low-Dimensional Quantum Systems

13.1 Mesoscopic Systems

13.2 The Landauer Conductance Formula

13.3 Properties of Quantum Dots

13.4 Disorder in Mesoscopic Systems

13.5 Kondo Effect in Quantum Dots

13.6 Graphene

13.7 Inventory of Recently Discovered Low-Dimensional Phenomena

14. Many-Body Theory

14.1 Second Quantization

14.2 Statistical Mechanics in Second Quantization

14.3 The Electron Gas

14.4 Mean-Field Theory

15. Density Functional Theory

15.1 The Hohenberg–Kohn Theorems

15.2 The Thomas–Fermi Approximation

15.3 The Kohn–Sham Equations

15.4 Spin DFT and Magnetic Systems

15.5 The Gap Problem in DFT

15.6 Time-Dependent DFT

15.7 DFT Computer Packages

A. Linear Algebra

A.1 Vector Spaces

A.2 Operators and Matrices

B. Some Ordinary Differential Equations

C. Vector Analysis

C.1 Scalar and Vector Products

C.2 Differential Operators

C.3 Divergence and Stokes Theorems

C.4 Curvilinear Coordinates

D. Fourier Analysis

D.1 Fourier Series

D.2 Fourier Integrals

D.3 Fourier Series and Integrals in Three-Space Dimensions

D.4 Fourier Integrals of Time-Dependent Functions

D.5 Convolution

D.6 Fourier Expansion of Operators

D.7 Fourier Transforms

D.8 FT for Solving Differential and Integral Equations

E. Symmetry and Group Theory

E.1 Group Theory Axioms

E.2 Group Multiplication Tables

E.3 Examples of Groups

E.4 Some Properties of Groups

E.5 Group Representations

Bibliography

Index