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 phenomenaThe fundamentals of quantum theory are provided within a modern perspective, with emphasis on applications to nanoscience and nanotechnology, and information-technologyOlder 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 nanotechnologyAs 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 todayThere are many excellent quantum mechanics books available, but none have the emphasis on nanotechnology and quantum information science that this book has

1 Introduction to Nanotechnology and Information-Technology

1.1 STM and AFM Microscopies

1.2 Molecular-Electronics

1.3 Quantum Dots, Wires and Wells, and Nanotubes

1.4 Bio-Nanotechnology

1.5 Information-Technology

2 The Formalism of Quantum Mechanics

2.1 Hilbert Space and Dirac Notation

2.2 The Postulates of Quantum Mechanics

2.2.1 The Measurement Problem

2.3 Hermitian and Anti-Hermitian operators

2.3.1 Compatible Operators and Degeneracy

2.3.2 Basis-State Expansions

2.4 The Uncertainty Principle

2.5 Mixed States: Density Matrix Formulation

2.5.1 Purity and von-Neumann Entropy

2.5.2 The Measurement Problem Revisited

2.6 Position and Momentum Representations

2.6.1 The Wigner Representation

2.7 Schrodinger and Heisenberg Representations

2.7.1 Interaction Representation

2.7.2 Harmonic Oscillator Raising and Lowering Operators

2.7.3 Coherent States and Squeezed States

2.8 The Correspondence Principle and the Classical Limit

2.9 Symmetry and Conservation Laws in Quantum Mechanics

2.9.1 Exchange Symmetry

2.9.2 Inversion Symmetry

2.9.3 Time-Reversal Symmetry

3 Angular Momentum and Spin

3.1 Angular Momentum in Quantum Mechanics

3.1.1 Angular Momentum Raising and Lowering Operators

3.1.2 Electron Spin: j = 1/2

3.1.3 Angular Momentum in Spherical Coordinates

3.1.4 Spherical Harmonics

3.2 Spin Angular Momentum

3.3 Spinors

3.3.1 Pauli Matrices

3.3.2 Rotation of Spinors

3.3.3 Spin-Orbitals

3.4 Electron in a Magnetic Field

3.4.1 Charged Particle in a Magnetic Field: Orbital Effects

3.4.2 Time-Reversal Properties of Spinors

3.5 Spin-Orbit Interaction

3.6 Hyperfine Interaction

3.6.1 Zeeman Splitting of Hyperfine States

3.7 Spin-Dipolar Interactions

3.8 Magnetic Resonance

3.8.1 The Rotating Wave Approximation

3.8.2 Spin Relaxation and The Bloch Equation

3.8.3 Chemical Shifts

3.8.4 Fourier Transform NMR

6 Quantum Information

6.1 Classical Information

6.1.1 Entropy and Information

6.1.2 Classical Bits and Gates

6.1.3 Classical Cryptography

6.1.4 Computational Complexity

6.2 Quantum Information and Processing

6.2.1 Qubits and Entanglement

6.2.2 Quantum Gates

6.2.3 No-Cloning Theorem

6.2.4 Dense Coding

6.2.5 Data Compression

6.2.6 Quantum Teleportation

6.2.7 Quantum Cryptography

6.2.8 Quantum Computing Despite Measurement

6.3 Quantum Computing

6.3.1 Deutsch and Deutsch-Jozsa Algorithms

6.3.2 The Grover Search Algorithm

6.3.3 Quantum Fourier Transform

6.3.4 Shor Factorization Algorithm

6.3.5 Quantum Simulation

6.4 Decoherence

6.5 Quantum Error Correction

6.6 Experimental Implementations

6.6.1 Ion Traps

6.6.2 Neutral Atoms in Optical Lattices

6.6.3 Cavity Based Quantum Computing

6.6.4 Nuclear Magnetic Resonance Systems

6.6.5 All-Optical Quantum Computers

6.6.6 Solid-State Qubits

6.7 The EPR Paradox

6.8 Bell’s Inequalities

7 Quantum Dynamics and Correlations

7.1 Two-Level Systems (Spin Systems)

7.1.1 Two-Level Dynamics (Spin Dynamics)

7.1.2 The Bloch Sphere Picture

7.1.3 Coupling to a Bath: Decoherence

7.1.4 Adiabatic limit: The Steady-State Approximation

7.1.5 Two or More Correlated Spins

7.1.6 N-Two-Level System Bloch Sphere

7.1.7 Ramsey Fringe Spectroscopy

7.2 Three-Level Systems

7.2.1 Two or More Three-Level Correlated Systems

7.2.2 Three-Level Dynamics

7.3 Continuous-Variable Systems

7.3.1 Wave Packet Dynamics

7.4 Time-Dependent Hamiltonians

7.5 Quantum Optimal Control Theory

8 Approximation Methods 371

8.1 Basis State Expansions

8.1.1 Time-Dependent Basis State Expansions

8.2 Semiclassical Approximations

8.2.1 The WKB Approximation

8.2.2 Semiclassical Expansion of Ehrenfest Theorem

8.2.3 Semiclassical Hamilton-Jacobi Expansion

8.3 Perturbation Theory

8.3.1 Non-degenerate Perturbation Theory

8.3.2 Perturbative Magnetic Field Effects

8.3.3 Perturbative Electric Field Effects

8.3.4 Degenerate Perturbation Theory

8.3.5 Time-Dependent Perturbation Theory

8.4 Dynamics in an Electromagnetic Field

8.4.1 Spontaneous and Stimulated Emission of Radiation

8.4.2 Electric Dipole and Multipole Radiation

8.4.3 Rayleigh, Raman and Brillouin Two-Photon Transitions

8.4.4 Decay Width

8.4.5 Doppler Shift

8.5 Exponential and Nonexponential Decay

8.6 The Variational Method

8.7 The Sudden Approximation

8.8 The Adiabatic Approximation

8.8.1 Chirped Pulse Adiabatic Passage

8.8.2 Stimulated Raman Adiabatic Passage

8.8.3 The Landau-Zener Problem

8.8.4 Generalized Displacements and Forces

8.8.5 Berry’s Phase

8.9 Linear Response and Susceptibilities

8.9.1 The Kubo Formula: Correlation Functions

8.9.2 Fluctuation-Dissipation Theorem

9 Identical Particles

9.1 Permutation Symmetry

9.1.1 The Symmetric Group

9.1.2 Young Tableaux

9.2 Exchange Symmetry

9.2.1 Symmetrization Postulate

9.3 Slater Determinants and Permanents

9.4 Simple Two-Electron States

9.5 Exchange Symmetry of Two Two-Level Systems

9.6 Exchange Symmetry of Many-Particle States

10 Electronic Properties of Solids

10.1 Free Electron Gas

10.1.1 Density of States in 2D and 1D systems

10.1.2 Fermi-Dirac Distribution Function

10.2 Elementary Theories of Conductivity

10.2.1 Drude Theory of Electron Conductivity

10.2.2 Thermal Conductivity of Metals

10.2.3 Sommerfeld Theory of Transport in Metals

10.3 Crystal Structure

10.3.1 Bravais Lattices and Crystal Systems

10.3.2 The Reciprocal Lattice

10.3.3 Quasicrystals

10.4 Electrons in a Periodic Potential

10.4.1 From Atomic Orbits to Band Structure

10.4.2 Band Structure and Electron Transport

10.4.3 Periodic Potential and Band Formation

10.4.4 Sinusoidal Potential: Mathieu Functions

10.4.5 Bloch Wave Functions and Energy Bands

10.4.6 Schr¬odinger Equation in Reciprocal Lattice Space

10.4.7 Tight-Binding Approximation

10.4.8 Wannier Functions

10.4.9 Electric Field Effects

10.5 Magnetic Field Effects

10.5.1 The Aharonov-Bohm Effect

10.5.2 Landau Levels

10.5.3 Periodic Potential and Magnetic Field

10.5.4 The Hall Effect and Magnetoresistance

10.5.5 de Haas-van Alphen and Shubnikov-de Haas Effects

10.5.6 The Quantum Hall Effect

10.5.7 Paramagnetism and Diamagnetism

10.6 Semiconductors

10.6.1 Semiconductor Band Structure

10.6.2 Density of Charge Carriers

10.6.3 Donor and Acceptor Impurities

10.6.4 p-n Junctions

10.6.5 Excitons

10.6.6 Low Energy Excitations

10.6.7 Spin-Orbit Coupling in Solid-State Physics

10.6.8 káp Perturbation Theory

10.6.9 Photon Induced Processes in Semiconductors

10.7 Insulators

10.7.1 On the Definition of the Gap

10.7.2 Nature of Band, Peirles and Anderson Insulators

10.7.3 Mott Insulators

10.8 Spintronics

11 Electronic Structure of Multi-Electron Systems

11.1 The Multi-Electron System Hamiltonian

11.2 Slater and Gaussian Type Atomic Orbitals

11.3 Term Symbols for Atoms

11.4 Two-Electron Systems

11.4.1 The Helium Atom

11.4.2 The Hartree Method: Helium

11.5 Hartree Approximation for Multi-Electron Systems

11.5.1 Koopmans’ Theorem

11.6 Hartree-Fock for Multi-electron Atoms

11.6.1 Hartree-Fock for Helium

11.7 Electronic Structure of Molecules

11.7.1 H+ Molecular Orbitals

11.7.2 The Hydrogen-Molecule

11.7.3 The H¬uckel Approximation

11.8 Hartree-Fock for Metals

11.9 Electron Correlation

11.9.1 Configuration Interaction

11.9.2 Moller-Plesset Many-Body Perturbation Theory

11.9.3 Coupled Cluster Method

11.10Multi-Electron Fine and Hyperfine Structure

12 Molecules

12.1 Molecular Orbitals and Group Theory

12.1.1 Character Tables and Mulliken Symbols

12.2 Diatomic Electronic States

12.2.1 Hund’s Coupling Cases

12.2.2 Hyperfine Interactions in Diatomic Molecules

12.3 The Born-Oppenheimer Approximation

12.3.1 Potential Energy Crossings and Pseudo-Crossings

12.3.2 Born-Oppenheimer Nuclear Derivative Coupling

12.3.3 The Hellman-Feynman Theorem

12.4 Rotational-Vibrational Structure

12.5 Electronic Optical Transition Selection Rules

12.6 The Franck-Condon Principle

13 Scattering Theory

13.1 Classical Scattering Theory

13.2 Quantum Treatment of Scattering

13.2.1 Time-Dependent Formulation

13.3 Stationary Scattering Theory

13.3.1 Cross-Sections

13.3.2 Two-Body Potential Scattering

13.3.3 From Wave Functions to Cross-Sections

13.3.4 Green’s Functions

13.4 Aspects of Formal Scattering Theory

13.4.1 The S Matrix and M¬oller Operators

13.5 Spherically Symmetric Potentials

13.5.1 Partial Wave Analysis

13.5.2 Phase Shift Analysis

13.5.3 Scattering from a Coulomb Potential

13.5.4 Scattering of Two Identical Particles

13.6 Resonance Scattering

13.6.1 Low Energy Cross-Sections: Influence of Bound-States

13.6.2 Resonance Cross-Sections

13.6.3 Feshbach Resonance

13.6.4 Fano Resonance

13.7 Approximation Methods

13.7.1 Born approximation

13.7.2 WKB Approximation

13.7.3 Variational Principle

13.7.4 Eikonal Approximation

13.8 Scattering of Particles with Internal Degrees of Freedom

13.8.1 Asymptotic States and Cross-Sections

13.8.2 The Multichannel S Matrix

13.8.3 Scattering from Two Potentials

13.8.4 Scattering of Particles with Spin

13.8.5 Inelastic Scattering and Scattering Reactions

13.8.6 Scattering from Identical Particles

13.9 Scattering in Low Dimensional Systems

13.9.1 Scattering in Two Dimensions

13.9.2 Scattering in One Dimension: The S matrix

13.9.3 Scattering in One Dimension: Anderson Localization

13.9.4 Scattering in Quasi-One-Dimensional Systems

13.9.5 The Landauer Conductance Formula

14 Many-Body Theory

14.1 Second Quantization

14.1.1 Construction of a basis

14.1.2 Mapping onto Fock Space

14.1.3 Creation and Annihilation operators

14.1.4 The Hamiltonian in Fock Space

14.1.5 Field Operators

14.1.6 Quantizing the Radiation Field: Photons

14.1.7 Quantizing Crystal Vibrations: Phonons

14.1.8 Systems with Two Kinds of Particles

14.2 Statistical Mechanics in Second Quantization

14.3 Mean-Field Theory

14.3.1 The Mean-Field Equations

14.3.2 The Hartree-Fock Approximation

14.4 Green’s Function

14.4.1 Representations

14.4.2 Green’s Functions in Many-Body Theory

14.4.3 Connection to Observables

14.4.4 The Spectral Function

14.4.5 Green’s Functions in Tunneling

14.5 Calculating Green’s Functions

14.5.1 Equations of Motion

14.5.2 Interaction Representation Expansion

14.5.3 Wick’s Theorem

14.5.4 Feynman Diagrams

14.5.5 Dyson Equation

14.5.6 Feynman Diagrams for Interacting Electrons

14.5.7 Higher Order Correlations

14.6 Imaginary Time Green’s Function

14.6.1 Matsubara Green’s function

14.7 Linear Response

14.8 Important Second Quantized Problems

14.8.1 Electrons and Phonons in a Crystal

14.8.2 Electrons and Photons

14.8.3 Anderson Impurity Model

14.8.4 The Kondo Effect

14.8.5 Spin Systems

14.8.6 Bose-Einstein Condensation

14.8.7 Landau Fermi Liquid Theory

14.8.8 Superconductivity and Superfluidity

15 Density Functional Theory

15.1 The Hohenberg-Kohn Theorems

15.2 The Thomas-Fermi Approximation

15.3 The Kohn-Sham Equations

15.3.1 Local Density Approximation

15.4 Spin DFT and Magnetic Systems

15.4.1 Spin DFT Local Density Approximation

15.5 Time-Dependent DFT

15.5.1 The Runge-Gross theorem

15.5.2 Time-Dependent Kohn-Sham equations

15.5.3 Adiabatic LDA Approximation

15.6 Temperature-Dependent DFT

15.6.1 Legendre Transforms and the Hohenberg-Kohn Theorem

15.6.2 The Finite T Kohn-Sham Equations

15.6.3 Finite T Bosonic Systems

15.6.4 The Grand Potential and the Free Energy

15.6.5 Solving the Effective non-interacting Reference System

16 Quantum Dots and Other Low-Dimensional Systems

16.1 Quantum Dots

16.1.1 Equilibrium Properties of Quantum Dots

16.1.2 Transport Properties of Quantum Dots

16.2 Quantum Wires

16.2.1 Nanotubes

16.3 Quantum Wells

16.3.1 Heterojunctions and Superlattices

16.4 Graphene

16.4.1 Charge Carriers in Graphene

16.4.2 Dirac Equation and its Relevance to Graphene

16.4.3 Tight Binding Model for Graphene

16.4.4 Continuum Theory

16.4.5 Landau Levels in Graphene

16.4.6 Potential Scattering in Graphene

16.5 Decoherence in Confined Systems

17 Decoherence and Dissipation

17.1 System-Bath Interactions

17.1.1 Decoherence Free Subspaces

17.2 Decohered and Dissipative Two-Level System

17.2.1 The Caldeira-Leggett Model

17.3 The Scattering Model of Decoherence and Localization

17.4 Master Equations

17.5 Quantum Fluctuations and Decoherence

17.6 Entanglement Decoherence

17.7 Weak Measurements and Non-Demolition Measurements

A Linear Algebra

A.1 Vector Spaces

A.1.1 Inner Product Spaces and Dirac Notation

A.2 Operators and Matrices

A.3 Determinants and Permanents

A.4 Antilinear and Antiunitary Operators

B Probability Theory

C Some Simple Ordinary Differential Equations

D Vector Analysis

D.1 Scalar and Vector Products

D.2 Differential Operators

D.3 Divergence and Stokes Theorems

D.4 Curvilinear Coordinates

E Fourier Analysis

E.1 Fourier Series

E.1.1 Fourier Series of Functions of a Discrete Variable

E.2 Fourier Integrals

E.3 Fourier Series and Integrals in Three Space Dimensions

E.3.1 3D Fourier Integrals

E.4 Fourier Integrals of Time-Dependent Functions

E.5 Convolution

E.6 Fourier Expansion of Operators

E.7 Fourier Transforms

E.8 FT for Solving Differential and Integral Equations

F Symmetry and Group Theory

F.1 Group Theory Axioms

F.2 Group Multiplication Tables

F.3 Examples of Groups

F.3.1 Point Groups

F.3.2 Space Groups

F.3.3 Continuous Groups

F.4 Some Properties of Groups

F.5 Group Representations

F.5.1 Irreducible Representations

F.5.2 Group Orthogonality Theorem

F.5.3 Characters and Character Tables

F.5.4 Constructing Irreducible Representations

F.6 Unitary Representations in Quantum States

F.7 SU(2), SU(1, 1), etc.

F.8 O(3), SO(3)