Note: Intro -- Nonequilibrium Green's Functions Approach to Inhomogeneous Systems -- Preface -- Acknowledgments -- Contents -- Acronyms -- Part I: Introduction -- Chapter 1: Quantum Many-Particle Systems out of Equilibrium -- 1.1 Overview on Computational Approaches -- 1.2 Many-Body Interactions in Inhomogeneous Quantum Systems -- 1.3 Correlations -- Part II: Theory -- Chapter 2: Nonequilibrium Green's Functions -- 2.1 Introduction -- 2.1.1 Keldysh Contour -- 2.1.2 One-Particle Nonequilibrium Green's Function -- 2.2 Equations of Motion -- 2.2.1 Keldysh-Kadanoff-Baym Equations -- 2.2.2 Equilibrium Limit. Dyson Equation -- 2.3 Many-Body Approximations -- 2.3.1 Requirements for a Conserving Scheme -- 2.3.2 Perturbation Expansions -- 2.4 Quantum Kinetic Equations for Single-Time Quantities -- 2.4.1 The Reconstruction Problem for the One-Particle Green's Function -- 2.4.2 The Generalized Kadanoff-Baym Ansatz -- Part III: Computational Methods -- Chapter 3: Representations of the Nonequilibrium Green's Function -- 3.1 Numerical Resources -- 3.1.1 Homogeneous Systems. A Brief Outline -- 3.1.2 Inhomogeneous Systems. Computer Memory as Limiting Factor -- 3.2 Grid versus Basis Representations for Inhomogeneous Systems -- 3.3 An Ef cient Solution: The Finite Element-Discrete Variable Representation -- 3.3.1 General Idea and Background -- 3.3.2 Construction of the FE-DVR Basis -- 3.3.3 Matrix Elements of Relevant Energies -- 3.3.4 First- and Second-Order Self-energies -- Chapter 4: Computation of Equilibrium States and Time-Propagation -- 4.1 Preparing the Initial State: Ground State or Equilibrium -- 4.1.1 Time or Frequency Space? -- 4.1.2 Solution of the Dyson Equation in tau-Space -- 4.2 Nonequilibrium -- 4.2.1 Two-Time Propagation Method -- 4.2.2 Parallelization Strategies -- 4.2.3 Single-Time Propagation using the GKBA. , Part IV: Applications for Inhomogeneous Systems -- Chapter 5: Lattice Systems -- 5.1 Overview -- 5.2 A Basic Example -- 5.2.1 Dynamics Following a Non-Perturbative Excitation -- 5.2.2 Absorption Spectrum in Second Born Approximation -- Chapter 6: Non-Lattice Systems -- 6.1 Small Atoms and Molecules. Ground State Properties and Response to External Fields -- 6.1.1 Model-Like Treatment -- 6.1.2 3D Atoms and Molecules -- 6.2 Few-Electron Quantum Dots and Wells -- 6.2.1 Correlation Effects in the Optical Absorption Spectra -- 6.2.2 Electronic Double Excitations from the Kadanoff-Baym Equations -- Chapter 7: Conclusion and Outlook -- 7.1 Summary -- 7.2 Prospects for Future Applications -- Appendix A: Second Quantization -- A.1 Symmetry of Many-Body States -- A.2 Occupation Number Representation -- A.3 Particle Creation and Annihilation in Fock Space -- A.4 General Form of Operators -- Appendix B: Perturbation Expansion. Supplements -- B.1 Derivative of a Contour-Ordered Product -- B.2 Equations for Sigma(1) and deltaG(1)/deltav(1) in Terms of deltaSigma(1)/deltav(1) -- References -- Index.

Note: Intro -- Preface -- Acknowledgements -- Contents -- 1 Operator Spaces -- 1.1 Operator Spaces -- 1.1.1 Completely Bounded and Completely Positive Maps -- 1.1.2 Operator Systems -- 1.1.3 Fundamental Factorisation of CB Maps -- 1.2 More on CB and CP Maps -- 1.3 Ruan's Theorem and Its Applications -- 1.3.1 Ruan's Theorem -- 1.3.2 Some Applications and Some Basic Facts -- Analogy with Banach Space Properties -- 1.3.3 min and max Operator Space Structures on a Banach Space -- 1.4 Tensor Products of Operator Spaces -- 1.4.1 Injective Tensor Product -- 1.4.2 Projective Tensor Product -- 1.4.3 General Remarks -- 1.4.4 A Passing Remark on the Haagerup Tensor Product -- 1.5 Tensor Products of C*-Algebras -- 1.5.1 min and max Tensor Products of C*-Algebras -- 1.5.2 Kirchberg's Theorem -- References -- 2 Entanglement in Bipartite Quantum States -- 2.1 Quantum States, Observables and Probabilities -- 2.2 Entanglement -- 2.2.1 Schmidt Decomposition -- 2.2.2 Unitary Bases, EPR States and Dense Coding -- 2.3 Schmidt Rank of Bipartite Entangled States -- 2.3.1 Subspaces of Minimal Schmidt Rank -- 2.4 Schmidt Number of Mixed States -- 2.4.1 Test for Schmidt Number k Using k-Positive Maps -- 2.4.2 Schmidt Number of Generalized Werner States -- References -- 3 Operator Systems -- 3.1 Theorems of Choi -- 3.1.1 Douglas Factorization -- 3.1.2 Choi-Kraus Representation and Choi Rank -- Example: Binary Case Quantum Error Detection/Correction -- 3.2 Quantum Error Correction -- 3.2.1 Applications of Choi's Theorems to Error Correction -- 3.2.2 Shor's Code: An Example -- 3.3 Matrix Ordered Systems and Operator Systems -- 3.3.1 Duals of Matrix Ordered Spaces -- 3.3.2 Choi-Effros Theorem -- 3.4 Tensor Products of Operator Systems -- 3.4.1 Minimal Tensor Product of Operator Systems -- 3.4.2 Maximal Tensor Product of Operator Systems. , An Example of a Nuclear Operator System that is Not a C*-Algebra HP11 -- 3.5 Graph Operator Systems -- 3.5.1 Dual of a Graph Operator System -- 3.6 Three More Operator System Tensor Products -- 3.6.1 The Commuting Tensor Product c. -- 3.6.2 The Tensor Products el and er. -- 3.6.3 Lattice of Operator System Tensor Products -- 3.7 Some Characterizations of Operator System Tensor Products -- 3.7.1 Exact Operator Systems -- 3.7.2 Weak Expectation Property (WEP) -- 3.7.3 Operator System Local Lifting Property (OSLLP) -- 3.7.4 Double Commutant Expectation Property (DCEP) -- 3.8 Operator System Tensor Products and the Conjectures of Kirchberg and Tsirelson -- 3.8.1 Special Operator Sub-systems of the Free Group C*-Algebras -- 3.8.2 Kirchberg's Conjecture -- 3.8.3 Quotient of an Operator System -- References -- 4 Quantum Information Theory -- 4.1 Zero-Error Communication Via Quantum Channels -- 4.1.1 Conditions for Zero-Error Quantum Communication -- 4.1.2 Zero-Error Capacity and Lovasz Function -- 4.2 Strong Subadditivity and Its Equality Case -- 4.2.1 Monotonicity of Relative Entropy: Petz Theorem -- 4.2.2 Structure of States that Saturate Strong Subadditivity -- 4.2.3 An Operator Algebraic Proof of the Koashi-ImotoTheorem -- 4.3 Norms on Quantum States and Channels -- 4.4 Matrix-Valued Random Variables -- 4.4.1 Matrix Tail Bounds -- 4.4.2 Destroying Correlations -- 4.4.3 State Merging -- References -- Further Reading -- Index.

Note: Intro -- Preface -- Contents -- 1 Equiangular Lines -- 1.1 Introduction -- 1.2 Real Lines -- 1.3 Complex Lines -- References -- 2 Optimal Quantum Measurements -- 2.1 Introduction -- 2.2 SIC Representations of Quantum States -- 2.3 Constructing SICs Using Groups -- References -- 3 Geometry and Information Theory for Qubits and Qutrits -- 3.1 Qubits -- 3.2 Qutrits -- 3.3 Coherence -- References -- 4 SICs and Bell Inequalities -- 4.1 Mermin's Three-Qubit Bell Inequality -- 4.2 The Hoggar SIC -- 4.3 Qubit Pairs and Twinned Tetrahedral SICs -- 4.4 Failure of Hidden Variables for Qutrits -- 4.5 Quantum Theory from Nonclassical Probability Meshing -- References -- 5 The Hoggar-Type SICs -- 5.1 Introduction -- 5.2 Simplifying the QBic Equation -- 5.3 Triple Products and Combinatorial Designs -- 5.4 The Twin of the Hoggar SIC -- 5.5 Combinatorial Designs from the Twin Hoggar SIC -- 5.6 Quantum-State Compatibility -- 5.7 From Pauli Operators to Real Equiangular Lines -- 5.8 Concluding Remarks -- References -- 6 Sporadic SICs and the Exceptional Lie Algebras -- 6.1 Root Systems and Lie Algebras -- 6.2 E6 -- 6.3 E8 -- 6.4 E7 -- 6.5 The Regular Icosahedron and Real-Vector-Space Quantum Theory -- 6.6 Open Puzzles Concerning Exceptional Objects -- References -- 7 Exercises -- References -- Appendix Index -- Index.

Note: Intro -- Preface -- Contents -- Notation -- Symbols -- Abbreviations -- Chapter 1 Introduction -- References -- Chapter 2 Review of Quantum Mechanics -- 2.1 Historical Background -- 2.2 Postulates of Quantum Mechanics -- 2.2.1 Quantum States -- 2.2.2 Operators -- 2.2.3 Measurements and Expectation Values -- 2.2.4 Schrödinger Equation -- 2.3 Spin -- 2.3.1 Spinors and Pauli Equation -- 2.3.2 Spin-Orbit Coupling -- References -- Chapter 3 Many-Body Systems -- 3.1 First Quantization -- 3.1.1 Indistinguishability -- 3.1.2 Slater Determinants and Permanents -- Fermions -- Bosons -- 3.1.3 Operators in the First Quantization Representation -- One-Body Operators -- Two-Body Operators -- 3.2 Second Quantization -- 3.2.1 Creation and Annihilation Operators -- Bosons -- Fermions -- 3.2.2 Operators in the Second Quantization Representation -- One-Body Operators -- Two-Body Operators -- 3.2.3 Basis Transformation -- 3.2.4 Field Operators -- 3.2.5 Quasi-particles and Collective Excitations -- 3.2.6 Harmonic Oscillator -- 3.2.7 Photons -- 3.2.8 Interaction with Photons -- References -- Chapter 4 Band Theory -- 4.1 Crystal Lattices -- 4.2 Electrons in Crystals -- 4.2.1 Bloch States -- 4.2.2 Tight-Binding Approximation -- 4.2.3 The Hubbard Model -- 4.3 Phonons -- 4.3.1 Phonon Interaction Potential -- Deformation Potential -- Polar Interaction -- 4.3.2 Scattering of Bloch States -- References -- Chapter 5 Statistical Mechanics -- 5.1 Historical Review -- 5.2 Basic Concepts -- 5.2.1 Macro and Microstates -- 5.2.2 Ergodicity -- 5.2.3 Classical and Quantum Statistics -- 5.3 Thermodynamics -- 5.3.1 The Laws of Thermodynamics -- 5.3.2 Closed Systems -- 5.3.3 Systems in Contact with a Heat Bath -- 5.3.4 Systems in Contact with a Heat and Particle Reservoir -- 5.3.5 Thermodynamic Potentials -- 5.3.6 Thermodynamic Equilibrium -- 5.3.7 Connection to Statistics. , 5.4 Statistical Ensembles -- 5.4.1 Micro-canonical Ensemble -- 5.4.2 Canonical Ensemble -- 5.4.3 Grand-Canonical Ensemble -- 5.5 Quantum Statistics -- 5.5.1 Density Matrix -- 5.5.2 Fermi-Dirac Statistics -- 5.5.3 Bose-Einstein Statistics -- 5.5.4 Maxwell-Boltzmann Statistics -- 5.6 Non-equilibrium Statistics -- 5.6.1 Boltzmann Transport Equation -- 5.6.2 Validity of the Boltzmann Transport Equation -- 5.6.3 Density Matrix -- 5.6.4 Wigner Representation -- 5.6.5 Green's Function -- References -- Chapter 6 Green's Function Formalism -- 6.1 Historical Review -- 6.2 Quantum Dynamics -- 6.2.1 Schrödinger Picture -- 6.2.2 Heisenberg Picture -- 6.2.3 Interaction Picture -- 6.2.4 The Evolution Operator -- 6.2.5 Imaginary Time Propagation -- 6.3 Equilibrium Green's Function -- 6.3.1 Zero Temperature Green's Function -- 6.3.2 Finite Temperature Green's Function -- 6.3.3 Matsubara Green's Function -- 6.4 Non-equilibrium Green's Functions -- 6.4.1 Non-equilibrium Ensemble Average -- 6.4.2 Contour-Ordered Green's Function -- 6.4.3 Keldysh Contour -- 6.4.4 Real-Time Formalism -- 6.4.5 Langreth Theorem -- 6.4.6 Non-interacting Fermions -- 6.4.7 Non-interacting Bosons -- 6.5 Perturbation Expansion of the Green's Function -- 6.5.1 Wick's Theorem -- 6.5.2 Feynman Diagrams -- 6.5.3 First-Order Perturbation Expansion -- 6.5.4 Dyson Equation -- 6.5.5 Electron-Electron Self-Energy -- 6.5.6 Electron-Phonon Self-Energy -- 6.6 Quantum Kinetic Equations -- 6.6.1 The Kadanoff-Baym Formulation -- 6.6.2 Keldysh Formulation -- 6.6.3 Steady-State Kinetic Equations -- 6.7 Variational Derivation of Self-Energies -- 6.7.1 Electron-Electron Interaction -- 6.7.2 Screened Interaction, Polarization, and Vertex Function -- 6.7.3 Electron-Phonon Interaction -- 6.7.4 The Phonon Green's Function -- 6.7.5 The Phonon Self-Energy -- 6.7.6 Approximation of the Self-Energy. , 6.8 Relation to Observables -- 6.8.1 Electron and Hole Density -- 6.8.2 Spectral Function and Local Density of States -- 6.8.3 Current Density -- References -- Chapter 7 Implementation -- 7.1 Basis Functions and Matrix Representation -- 7.1.1 Free Transverse-Direction -- 7.1.2 Real-Space Representation -- 7.1.3 Coupled Mode-Space Approach -- 7.1.4 Decoupled Mode-Space -- 7.2 Contacts -- 7.2.1 Matrix Truncation -- 7.2.2 Surface Green's Function -- 7.2.3 Sancho-Rubio Iterative Method -- 7.2.4 Contact Self-Energies -- 7.2.5 Wide-Band Limit -- 7.3 Scattering Self-Energies -- 7.3.1 Electron-Phonon Scattering -- 7.3.2 Acoustic Phonon -- 7.3.3 Optical Phonons -- 7.3.4 Polar Optical Phonons -- 7.4 Recursive Method for Calculating Green's Functions -- 7.4.1 Retarded and Advanced Green's Functions -- 7.4.2 Lesser and Greater Green's Functions -- 7.5 Evaluation of Observables -- 7.5.1 Carrier Concentration -- 7.5.2 Current Density -- 7.5.3 Transmission Probability -- 7.6 Selection of the Energy Grid -- 7.6.1 Confined States -- 7.6.2 Non-adaptive Energy Grid -- 7.6.3 Adaptive Energy Grid -- 7.7 Self-Consistent Simulations -- 7.7.1 Self-Consistent Iteration Scheme -- 7.7.2 Convergence of the Self-Consistent Simulations -- References -- Chapter 8 Applications -- 8.1 Introduction -- 8.2 Electronic Transport -- 8.2.1 Transport Models -- 8.2.2 Line-Edge Roughness -- 8.2.3 Substrate Corrugation -- 8.3 Spin Transport -- 8.3.1 Multi-orbital Model -- 8.3.2 Transport Model -- 8.3.3 Results -- 8.4 Phonon Transport -- 8.4.1 Phonon Bandstructure -- 8.4.2 Phonon Green's Function -- 8.4.3 Phonon Thermal Conductivity -- 8.4.4 Ballistic Phonon Transport -- 8.5 Graphene-Based Tunneling Transistors -- 8.5.1 Modeling -- 8.5.2 Self-Consistent Potential -- 8.5.3 Device Characteristics -- 8.6 CNT and GNR-Based Photodetectors -- 8.6.1 Electron-Photon Self-Energy. , 8.6.2 Quantum Efficiency -- References -- Index.

Note: Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- Foreword to the First Edition -- Preface to the Second Edition -- Preface to the First Edition -- Introduction -- 1: Overview of Bohmian Mechanics -- 1.1 Historical Development of Bohmian Mechanics -- 1.1.1 Particles and Waves -- 1.1.2 Origins of the Quantum Theory -- 1.1.3 "Wave or Particle?" vs. "Wave and Particle" -- 1.1.4 Louis de Broglie and the Fifth Solvay Conference -- 1.1.5 Albert Einstein and Locality -- 1.1.6 David Bohm and Why the "Impossibility Proofs" were Wrong? -- 1.1.7 John Bell and Nonlocality -- 1.1.8 Quantum Hydrodynamics -- 1.1.9 Is Bohmian Mechanics a Useful Theory? -- 1.2 Bohmian Mechanics for a Single Particle -- 1.2.1 Preliminary Discussions -- 1.2.2 Creating a Wave Equation for Classical Mechanics -- 1.2.2.1 Newton's second law -- 1.2.2.2 Hamilton's principle -- 1.2.2.3 Lagrange's equation -- 1.2.2.4 Equation for an (infinite) ensemble of trajectories -- 1.2.2.5 Classical Hamilton-Jacobi equation -- 1.2.2.6 Local continuity equation for an (infinite) ensemble of classical particles -- 1.2.2.7 Classical wave equation -- 1.2.3 Trajectories for Quantum Systems -- 1.2.3.1 Schrödinger equation -- 1.2.3.2 Local conservation law for an (infinite) ensemble of quantum trajectories -- 1.2.3.3 Velocity of Bohmian particles -- 1.2.3.4 Quantum Hamilton-Jacobi equation -- 1.2.3.5 A quantum Newton-like equation -- 1.2.4 Similarities and Differences between Classical and Quantum Mechanics -- 1.2.5 Feynman Paths -- 1.2.6 Basic Postulates for a Single-Particle -- 1.3 Bohmian Mechanics for Many-Particle Systems -- 1.3.1 Preliminary Discussions: The Many Body Problem -- 1.3.2 Many-Particle Quantum Trajectories -- 1.3.2.1 Many-particle continuity equation -- 1.3.2.2 Many-particle quantum Hamilton-Jacobi equation. , 1.3.3 Factorizability, Entanglement, and Correlations -- 1.3.4 Spin and Identical Particles -- 1.3.4.1 Single-particle with s = 1/2 -- 1.3.4.2 Many-particle system with s = 1/2 particles -- 1.3.5 Basic Postulates for Many-Particle Systems -- 1.3.6 The Conditional Wave Function: Many-Particle Bohmian Trajectories without the Many-Particle Wave Function -- 1.3.6.1 Single-particle pseudo-Schrödinger equation for many-particle systems -- 1.3.6.2 Example: Application in factorizable many-particle systems -- 1.3.6.3 Example: Application in interacting many-particle systems without exchange interaction -- 1.3.6.4 Example: Application in interacting many-particle systems with exchange interaction -- 1.4 Bohmian Explanation of the Measurement Process -- 1.4.1 The Measurement Problem -- 1.4.1.1 The orthodox measurement process -- 1.4.1.2 The Bohmian measurement process -- 1.4.2 Theory of the Bohmian Measurement Process -- 1.4.2.1 Example: Bohmian measurement of the momentum -- 1.4.2.2 Example: Sequential Bohmian measurement of the transmitted and reflected particles -- 1.4.3 The Evaluation of a Mean Value in Terms of Hermitian Operators -- 1.4.3.1 Why Hermitian operators in Bohmian mechanics? -- 1.4.3.2 Mean value from the list of outcomes and their probabilities -- 1.4.3.3 Mean value from the wave function and the operators -- 1.4.3.4 Mean value from Bohmian mechanics in the position representation -- 1.4.3.5 Mean value from Bohmian trajectories -- 1.4.3.6 On the meaning of local Bohmian operators AB(x) -- 1.5 Concluding Remarks -- 1.6 Problems and Solutions -- A.1 Appendix: Numerical Algorithms for the Computation of Bohmian Mechanics -- A.1.1 Analytical Computation of Bohmian Trajectories -- A.1.1.1 Time-dependent Schrödinger equation for a 1D space (TDSE1D-BT) with an explicit method. , A.1.1.2 Time-independent Schrödinger equation for a 1D space (TISE1D) with an implicit (matrix inversion) method -- A.1.1.3 Time-independent Schrödinger equation for a 1D space (TISE1D) with an explicit method -- A.1.2 Synthetic Computation of Bohmian Trajectories -- A.1.2.1 Time-dependent quantum Hamilton-Jacobi equations (TDQHJE1D) with an implicit (Newton-like fixed Eulerian mesh) method -- A.1.2.2 Time-dependent quantum Hamilton-Jacobi equations (TDQHJE1D) with an explicit (Lagrangian mesh) method -- A.1.3 More Elaborated Algorithms -- 2: Hydrogen Photoionization with Strong Lasers -- 2.1 Introduction -- 2.1.1 A Brief Overview of Photoionization -- 2.1.2 The Computational Problem of Photoionization -- 2.1.3 Photoionization with Bohmian Trajectories -- 2.2 One-Dimensional Photoionization of Hydrogen -- 2.2.1 The Physical Model -- 2.2.2 Harmonic Generation -- 2.2.3 Above Threshold Ionization -- 2.3 Hydrogen Photoionization with Beams Carrying Orbital Angular Momentum -- 2.3.1 Physical System -- 2.3.2 Bohmian Equations in an Electromagnetic Field -- 2.3.3 Selection Rules -- 2.3.4 Numerical Simulations -- 2.3.4.1 Gaussian pulses -- 2.3.4.2 Laguerre-Gaussian pulses -- 2.4 Conclusions -- 3: Atomtronics: Coherent Control of Atomic Flow via Adiabatic Passage -- 3.1 Introduction -- 3.1.1 Atomtronics -- 3.1.2 Three-Level Atom Optics -- 3.1.3 Adiabatic Transport with Trajectories -- 3.2 Physical System: Neutral Atoms in Optical Microtraps -- 3.2.1 One-Dimensional Hamiltonian -- 3.3 Adiabatic Transport of a Single Atom -- 3.3.1 The Matter Wave STIRAP Paradox with Bohmian Trajectories -- 3.3.2 Velocities and Accelerations of Bohmian Trajectories -- 3.4 Adiabatic Transport of a Single Hole -- 3.4.1 Hole Transfer as an Array-Cleaning Technique -- 3.4.2 Adiabatic Transport of a Hole in an Array of Three Traps -- 3.4.2.1 Three-level approximation description. , 3.4.2.2 Numerical simulations -- 3.4.3 Hole Transport Fidelity -- 3.4.4 Bohmian Trajectories for the Hole Transport -- 3.4.5 Atomtronics with Holes -- 3.4.5.1 Single-hole diode -- 3.4.5.2 Single-hole transistor -- 3.5 Adiabatic Transport of a Bose-Einstein Condensate -- 3.5.1 Madelung Hydrodynamic Formulation -- 3.5.2 Numerical Simulations -- 3.6 Conclusions -- 4: Bohmian Pathways into Chemistry: A Brief Overview -- 4.1 Introduction -- 4.2 Approaching Molecular Systems at Different Levels -- 4.2.1 The Born-Oppenheimer Approximation -- 4.2.2 Electronic Configuration -- 4.2.3 Dynamics of "Small" Molecular Systems -- 4.2.4 Statistical Approach to Large (Complex) Molecular Systems -- 4.3 Bohmian Mechanics -- 4.3.1 Fundamentals -- 4.3.2 Nonlocality and Entanglement -- 4.3.3 Weak Values and Equations of Change -- 4.4 Applications -- 4.4.1 Time-Dependent DFT: The Quantum Hydrodynamic Route -- 4.4.2 Bound System Dynamics: Chemical Reactivity -- 4.4.3 Scattering Dynamics: Young's Two-Slit Experiment -- 4.4.4 Effective Dynamical Treatments: Decoherence and Reduced Bohmian Trajectories -- 4.4.5 Pathways to Complex Molecular Systems: Mixed Bohmian-Classical Mechanics -- 4.5 Concluding Remarks -- 5: Adaptive Quantum Monte Carlo Approach States for High-Dimensional Systems -- 5.1 Introduction -- 5.2 Mixture Modeling Approach -- 5.2.1 Motivation for a Trajectory-Based Approach -- 5.2.1.1 Bohmian interpretation -- 5.2.1.2 Quantum hydrodynamic trajectories -- 5.2.1.3 Computational considerations -- 5.2.2 Density Estimation -- 5.2.2.1 The mixture model -- 5.2.2.2 Expectation maximization -- 5.2.3 Computational Results -- 5.2.3.1 Bivariate distribution with multiple nonseparable Gaussian components -- 5.2.4 The Ground State of Methyl Iodide -- 5.3 Quantum Effects in Atomic Clusters at Finite Temperature -- 5.4 Quantum Structures at Zero and Finite Temperature. , 5.4.1 Zero Temperature Theory -- 5.4.2 Finite Temperature Theory -- 5.4.2.1 Computational approach: The mixture model -- 5.4.2.2 Computational approach: Equations of motion for the sample points -- 5.4.3 Computational Studies -- 5.4.3.1 Zero temperature results -- 5.4.3.2 Finite temperature results -- 5.5 Overcoming the Node Problem -- 5.5.1 Supersymmetric Quantum Mechanics -- 5.5.2 Implementation of SUSY QM in an Adaptive Monte Carlo Scheme -- 5.5.3 Test Case: Tunneling in a Double-Well Potential -- 5.5.4 Extension to Higher Dimensions -- 5.5.4.1 Discussion -- 5.6 Summary -- 6: Nanoelectronics: Quantum Electron Transport -- 6.1 Introduction: From Electronics to Nanoelectronics -- 6.2 Evaluation of the Electrical Current and Its Fluctuations -- 6.2.1 Bohmian Measurement of the Current as a Function of the Particle Positions -- 6.2.1.1 Relationship between current in the ammeter Iammeter, g(t) and the current in the device-active region Ig(t) -- 6.2.1.2 Relationship between the current on the device-active region Ig(t) and the Bohmian trajectories {r1,g[t], . . . , rMP,g[t]} -- 6.2.1.3 Reducing the number of degrees of freedom of the whole circuit -- 6.2.2 Practical Computation of DC, AC, and Transient Currents -- 6.2.3 Practical Computation of Current Fluctuations and Higher Moments -- 6.2.3.1 Thermal and shot noise -- 6.2.3.2 Practical computation of current fluctuations -- 6.3 Solving Many-Particle Systems with Bohmian Trajectories -- 6.3.1 Coulomb Interaction Among Electrons -- 6.3.2 Exchange and Coulomb Interaction Among Electrons -- 6.3.2.1 Algorithm for spinless electrons -- 6.3.2.2 Algorithm for electrons with spins in arbitrary directions -- 6.4 Dissipation with Bohmian Mechanics -- 6.4.1 Parabolic Band Structures: Pseudo Schrödinger Equation -- 6.4.2 Linear Band Structures: Pseudo Dirac Equation -- 6.5 The BITLLES Simulator. , 6.5.1 Overall Charge Neutrality and Current Conservation.

Note: Intro -- Abstract -- Contents -- 1 Introduction -- 2 Mathematical Preliminaries -- 2.1 Basic Concepts of Vector Spaces -- 2.1.1 Minkowski Functionals -- 2.1.2 The Hahn-Banach Theorem -- 2.1.3 Locally Compact Spaces -- 2.1.4 Seminorms and Local Convexity -- 2.2 Vector Spaces With Order Unit -- 2.3 Base-norm Spaces -- 2.4 Functional Representation -- 2.5 Archimedeanization and Categories -- 2.6 Tensor Products -- 2.6.1 Tensor Products of Vector Spaces -- 2.6.2 Tensor Products of Order Unit Spaces -- 2.7 C*-algebras -- 2.8 Completely Positive Maps -- 2.9 Convex Polytopes -- 3 Generalized probabilistic theories -- 3.1 Preparation and Measurements -- 3.2 Attributes of GPTs -- 3.3 Examples of GPTs -- 4 Sections and Subsystems -- 4.1 Sections of the Effect Cone -- 4.2 Sections of the State Space -- 4.3 Spekkens' Toy Model -- 5 Two-Sections of Quantum Mechanics -- 5.1 Classification of Two Dimensional Sections -- 5.2 Geometric Evolution of State Spaces -- 6 Conclusion -- Bibliography.

Note: Cover -- Contents -- Abbreviations -- 1 Introduction -- 1.1 The third quantum revolution -- 1.2 Cold atoms from a historical perspective -- 1.3 Cold atoms and the challenges of condensed matter physics -- 1.4 Plan of the book -- 2 Statistical physics of condensed matter: basic concepts -- 2.1 Classical phase transitions -- 2.2 Bose-Einstein condensation in non-interacting systems -- 2.3 Quantum phase transitions -- 2.4 One-dimensional systems -- 2.5 Two-dimensional systems -- 3 Ultracold gases in optical lattices: basic concepts -- 3.1 Optical potentials -- 3.2 Control of parameters in cold atom systems -- 3.3 Non-interacting particles in periodic lattices: band structure -- 3.4 Bose-Einstein condensates in optical lattices: weak interacting limit -- 3.5 From weakly interacting to strongly correlated regimes -- 4 Quantum simulators of condensed matter -- 4.1 Quantum simulators -- 4.2 Hubbard models -- 4.3 Spin models and quantum magnetism -- 5 Bose-Hubbard models: methods of treatment -- 5.1 Introduction -- 5.2 Weak interactions limit: the Bogoliubov approach -- 5.3 Strong interactions limit: strong coupling expansion -- 5.4 Perturbative mean-field approach -- 5.5 Gutzwiller approach -- 5.6 Exact diagonalization and the Lanczos method -- 5.7 Quantum Monte Carlo: path integral and worm algorithms -- 5.8 Phase-space methods -- 5.9 Analytic one-dimensional methods -- 5.10 Renormalization approaches in one dimension: DMRG and MPS -- 5.11 Renormalization approaches in two dimension: PEPS, MERA, and TNS -- 6 Fermi and Fermi-Bose Hubbard models: methods of treatment -- 6.1 Introduction -- 6.2 Fermi Hubbard model and BCS theory -- 6.3 Balanced BCS-BEC crossover -- 6.4 Mean-field description of imbalanced BCS-BEC crossover -- 6.5 Fermi Hubbard model and strongly correlated fermions -- 6.6 Hubbard models and effective Hamiltonians. , 6.7 Fermi-Bose Hubbard models -- 7 Ultracold spinor atomic gases -- 7.1 Introduction -- 7.2 Spinor interactions -- 7.3 Spinor Bose-Einstein condensates: mean-field phases -- 7.4 Spin textures and topological defects -- 7.5 Bosonic spinor gases in optical lattices -- 7.6 Spinor Fermi gases -- 8 Ultracold dipolar gases -- 8.1 Introduction -- 8.2 Properties of dipole-dipole interaction -- 8.3 Ultracold dipolar systems -- 8.4 Ultracold trapped dipolar gases -- 8.5 Dipolar gas in a lattice: extended Hubbard models -- 8.6 Dipolar bosons in a 2D optical lattice -- 8.7 Quantum Monte Carlo studies of dipolar gases -- 8.8 Further dipole effects -- 9 Disordered ultracold atomic gases -- 9.1 Introduction -- 9.2 Disorder in condensed matter -- 9.3 Realization of disorder in ultracold atomic gases -- 9.4 Disordered Bose-Einstein condensates -- 9.5 Disordered ultracold fermionic systems -- 9.6 Disordered ultracold Bose-Fermi and Bose-Bose mixtures -- 9.7 Spin glasses -- 9.8 Disorder-induced order -- 10 Frustrated ultracold atom systems -- 10.1 Introduction -- 10.2 Quantum antiferromagnets -- 10.3 Physics of frustrated quantum antiferromagnets -- 10.4 Realization of frustrated models with ultracold atoms -- 11 Ultracold atomic gases in 'artificial' gauge fields -- 11.1 Introduction -- 11.2 Ultracold atoms in rapidly rotating microtraps -- 11.3 Gauge symmetry in the lattice -- 11.4 Lattice gases in 'artificial' Abelian gauge fields -- 11.5 Lattice gases in 'artificial' non-Abelian gauge fields -- 11.6 Integer quantum Hall effect and emergence of Dirac fermions -- 11.7 Fractional quantum Hall effect in non-Abelian fields -- 11.8 Ultracold gases and lattice gauge theories -- 11.9 Generation of 'artificial' gauge fields -- 12 Many-body physics from a quantum information perspective -- 12.1 Introduction -- 12.2 Crash course on quantum information. , 12.3 Quantum phase transitions and entanglement -- 12.4 Area laws -- 12.5 The world according to tensor networks -- 13 Quantum information with lattice gases -- 13.1 Introduction -- 13.2 Quantum circuit model in optical lattices -- 13.3 One-way quantum computer with lattice gases -- 13.4 Topological quantum computing in optical lattices -- 13.5 Distributed quantum information -- 14 Detection of quantum systems realized with ultracold atoms -- 14.1 Introduction -- 14.2 Time of flight: first-order correlations -- 14.3 Time of flight and noise correlations: higher-order correlations -- 14.4 Bragg spectroscopy -- 14.5 Optical Bragg diffraction -- 14.6 Single-atom detectors -- 14.7 Quantum polarization spectroscopy -- 15 Perspectives: beyond standard optical lattices -- 15.1 Introduction -- 15.2 Beyond standard optical lattices: new trends -- 15.3 Standard optical lattices: what's new? -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X.

Note: Cover -- Title Page -- Copyright Page -- Dedication -- Preface -- Contents -- Notations & -- Fundamental Constants -- 1 Basic Concepts & -- Formulation -- 1.1 Introduction -- 1.2 Measurements -- 1.3 Basic Postulates -- (i) Postulate 1. State of the System -- (ii) Postulate 2. Superposition of States -- (iii) Postulate 3. Operators for Dynamical Variables -- (iv) Postulate 4. Eigenvalues and Eigenvectors of Operators -- (v) Postulate 5. Hermitian Operator -- (vi) Postulate 6. Basis Vectors and Completeness Condition -- (vii) Expectation Value -- 2 Representation Theory -- 2.1 Elements of Representation Theory -- (i) The Energy Representation, called the E-representatrion -- (ii) Operators as Matrices -- 2.2 Change in Representation and Unitary Transformation -- 2.3 Commuting Observables -- 2.4 Uncertainty Relation -- 3 Position & -- Momentum Operators -- 3.1 Position Operator and its Eigenkets -- 3.2 Spatial Translation and Momentum Operator -- 3.3 Momentum Operator in Position Basis -- 3.4 Momentum Wavefunction -- 3.5 Gaussian as Minimum Uncertainty Wave Packet -- 3.6 Extension to Three Dimension -- 4 Time Evolution of Quantum Systems -- 4.1 Time Evolution Operator -- 4.2 The Schrödinger Equation of Motion -- 4.3 Time Dependence of Expectation Values: Ehrenfest Theorem -- 4.4 The Schrödinger and Heisenberg Pictures -- 4.5 The Heisenberg Equation of Motion -- 4.6 Operator Form of the Hamiltonian: Classical Analogue -- 4.7 Time Dependence of the Base Kets -- 5 Propagators and Feynman Path Integral -- 5.1 Propagators -- 5.2 Feynman's Path -- 6 Application in One Dimension -- 6.1 Free Particle -- 6.2 Rectangular Potential Well -- (i) Bound State Solution -- (ii) Unbound State Solution -- 6.3 Rectangular Potential Barrier -- 6.4 Delta Function Potential -- 6.5 Oscillator Problem by Schrödinger Method. , 6.6 Linear Harmonic Oscillator by Operator Method -- (i) Time Evolution of Oscillator -- (ii) Coherent State -- 6.7 Periodic Potential -- 7 Rotation and Angular Momentum -- 7.1 Introduction -- 7.2 Rotation in Three Dimension -- 7.3 Rotation of System Kets -- 7.4 Eigenvalue and Eigenvectors of Angular Momentum -- 7.5 Matrix Representation of Angular Momentum Operator -- 7.6 Orbital Angular Momentum -- 8 Spin Angular Momentum -- 8.1 The Stern Gerlach Experiment -- 8.2 Matrix Representation of Spin -- 8.3 Finite Rotations in Spin-½ Space -- 8.4 Pauli Two Component Spinor Formalism -- 9 Addition of Angular Momenta -- 9.1 Addition of Two Angular Momenta Ĵ[sub(1)] and Ĵ[sub(2)] -- 9.2 Addition of Orbital Angular Momentum and Spin of a Particle -- 9.3 Addition of Two Spins -- 10 Applications II -- 10.1 Hydrogen Atom -- 10.2 Charged Particle in Magnetic Field -- (i) The Landau Levels -- (ii) The Aharanov Bohm Effect -- 11 Symmetry in Quantum Mechanics -- 11.1 Symmetry Principle and Conservation Laws -- (i) Symmetry and Degeneracy -- 11.2 Space Reflection or Parity Operation -- 11.3 Time Reversal Symmetry -- 12 Approximate Methods -- 12.1 Semiclassical Aprroximation or WKB Method -- 12.2 Rayleigh Schrödinger Perturbation -- (i) Non-Degenerate Case -- (ii) Perturbation Calculations for Degenerate Energy Levels -- 12.3 The Variational Method -- 13 Methods for Time Dependent Problems -- 13.1 Time Dependent Perturbation -- 13.2 Harmonic Perturbation -- (i) Fermi's Golden Rule -- (ii) Ionization of Hydrogen Atom -- 13.3 Adiabatic Approximation -- 13.4 The Sudden Approximation -- 14 Scattering Theory I -- 14.1 Scattering Experiments: Cross Section -- 14.2 Potential Scattering -- 14.3 The Method of Partial Waves -- 14.4 The Optical Theorem -- 15 Scattering Theory II -- 15.1 The Lippmann Schwinger Equation -- 15.2 The Born Approximation. , 15.3 The Higher Order Born Approximation -- 16 Relativistic Wave Equations -- 16.1 Introduction -- 16.2 The Klein Gordon Equation -- 16.3 The Dirac Relativistic Equation -- 16.4 Conclusion -- A: Appendix -- A.1 Expansion in a Series of Orthonormal Functions -- A.2 Fourier Series -- A.3 Fourier Transforms -- A.4 The Dirac Delta Function -- Bibliography -- Index.

Note: Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Introduction -- 1.1 Scope -- 1.2 Conventions and notation -- 1.3 The independent-particle approximation -- 1.4 Electron correlation -- 1.5 Configuration interaction -- 1.6 Motivation -- 1.7 Extensivity -- 1.8 Disconnected clusters and extensivity -- 2 Formal perturbation theory -- 2.1 Background -- 2.2 Classical derivation of Rayleigh-Schrodinger perturbation theory -- 2.2.1 The perturbation Ansatz -- 2.2.2 Indeterminacy of the solution -- 2.2.3 Energy expressions -- 2.2.4 Order-by-order expansion -- 2.2.5 Expansion in zero-order functions -- 2.2.6 Wigner's 2n + 1 rule -- 2.2.7 The Hylleraas variation principle for the first-order wave function -- 2.3 Projection operators -- 2.4 General derivation of formal time-independent perturbation theories -- 2.4.1 General formalism -- 2.4.2 Brillouin-Wigner perturbation theory -- 2.4.3 Demonstration of non-extensivity of finite-order BWPT -- 2.4.4 Formal Rayleigh-Schrodinger perturbation theory -- 2.4.5 The general (non-diagonal) case -- 2.4.6 Bracketing procedure for RSPT -- 2.4.7 Summary of formal RSPT results -- 2.4.8 Extensivity of Rayleigh-Schrodinger perturbation theory -- 2.5 Similarity transformation derivation of the formal perturbation equations and quasidegenerate PT -- 2.6 Other approaches -- 3 Second quantization -- 3.1 Background -- 3.2 Creation and annihilation operators -- 3.2.1 Definitions -- 3.2.2 Anticommutation relations -- 3.2.3 Representation of operators -- 3.2.4 Invariance under unitary transformations -- 3.3 Normal products and Wick's theorem -- 3.3.1 Normal products of operators -- 3.3.2 Contractions -- 3.3.3 Time-independent Wick's theorem -- 3.3.4 Outline of proof of Wick's theorem -- 3.4 Particle-hole formulation -- 3.4.1 The reference state. , 3.4.2 Normal products and Wick's theorem relative to the Fermi vacuum -- 3.5 Partitioning of the Hamiltonian -- 3.6 Normal-product form of the quantum-mechanical operators -- 3.6.1 One-electron operators -- 3.6.2 Two-electron operators -- 3.6.3 The normal-product Hamiltonian -- 3.7 Generalized time-independent Wick's theorem -- 3.8 Evaluation of matrix elements -- 4 Diagrammatic notation -- 4.1 Time ordering -- 4.2 Slater determinants -- 4.3 One-particle operators -- 4.3.1 Representation of one-particle operators and contractions -- 4.3.2 Rules of interpretation -- 4.3.3 The complete one-particle operator -- 4.3.4 Products of one-particle operators -- 4.3.5 Phase factors -- 4.4 Two-particle operators -- 4.4.1 Goldstone diagrams for a two-particle operator -- 4.4.2 Vacuum expectation values of products of two-particle operators -- 4.4.3 Hugenholtz diagrams -- 4.4.4 Antisymmetrized Goldstone diagrams -- 4.4.5 Representation of operators not in normal-product form -- 4.4.6 The RSPT perturbation operator -- 5 Diagrammatic expansions for perturbation theory -- 5.1 Resolvent operator and denominators -- 5.2 First-order energy -- 5.3 Second-order energy -- 5.4 Third-order energy -- 5.5 Conjugate diagrams -- 5.6 Wave-function diagrams -- 5.6.1 First-order wave function -- 5.6.2 Second-order wave function -- 5.7 Fourth-order energy -- 5.7.1 Energy formula -- 5.7.2 Diagrams for E(4) in the canonical HF case -- 5.7.3 Non-HF diagrams for E(4) -- 5.7.4 Cancellation of the unlinked diagrams in E(4) -- 5.7.5 Role of the EPV terms -- 5.8 Linked-diagram theorem -- 5.9 Numerical example -- 5.10 Unlinked diagrams and extensivity -- 5.10.1 Extensivity implications -- 5.10.2 The dependence of diagrams -- 5.10.3 Relationship to configuration interaction (CI) -- 6 Proof of the linked-diagram theorem -- 6.1 The factorization theorem -- 6.2 The linked-diagram theorem. , 7 Computational aspects of MBPT -- 7.1 Techniques of diagram summation -- 7.2 Factorization of fourth-order quadruple-excitation diagrams -- 7.3 Spin summations -- 8 Open-shell and quasidegenerate perturbation theory -- 8.1 Formal quasidegenerate perturbation theory (QDPT) -- 8.2 The Fermi vacuum and the model states -- 8.3 Normal-product form of the generalized Bloch equations -- 8.4 Diagrammatic notation for QDPT -- 8.5 Schematic representation of the generalized Bloch equation -- 8.6 Level-shift and wave-operator diagrams -- 8.6.1 Zero and first order -- 8.6.2 Second-order level-shift operator -- 8.6.3 Second-order wave operator -- 8.6.4 Weight factors and phases -- 8.6.5 Folded diagrams -- 8.6.6 Third-order level-shift operator -- 8.6.7 Third-order wave operator -- 8.7 Incomplete model space -- 8.7.1 The Hose-Kaldor approach -- 8.7.2 The one-electron interaction -- 8.7.3 First-order diagrams -- 8.7.4 Second-order diagrams -- 8.7.5 Third-order level-shift diagrams -- 9 Foundations of coupled-cluster theory -- 9.1 Coupled-cluster theory for noninteracting He atoms -- 9.2 The coupled-cluster wave function -- 9.2.1 The exponential Ansatz and extensivity -- 9.2.2 The cluster operators -- 9.3 The coupled-cluster doubles (CCD) equations -- 9.3.1 Coupled-cluster doubles equations: configuration-space derivation -- 9.3.2 Coupled-cluster doubles equations: algebraic derivation -- 9.4 Exponential Ansatz and the linked-diagram theorem of MBPT -- 9.5 Diagrammatic derivation of the CCD equations -- 10 Systematic derivation of the coupled-cluster equations -- 10.1 The connected form of the CC equations -- 10.2 The general form of CC diagrams -- 10.3 Systematic generation of CC diagrams -- 10.4 The coupled-cluster singles and doubles (CCSD) equations -- 10.5 Coupled-cluster singles, doubles and triples (CCSDT) equations. , 10.6 Coupled-cluster singles, doubles, triples and quadruples (CCSDTQ) equations -- 10.7 Coupled-cluster effective-Hamiltonian diagrams -- 10.8 Results of various CC methods compared with full CI -- 11 Calculation of properties in coupled-cluster theory -- 11.1 Expectation value for a CC wave function -- 11.2 Reduced density matrices -- 11.3 The response treatment of properties -- 11.4 The CC energy functional -- 11.5 The Λ equations -- 11.6 Effective-Hamiltonian form of the… -- 11.7 Response treatment of the density matrices -- 11.8 The perturbed reference function -- 11.9 The CC correlation-energy derivative -- 12 Additional aspects of coupled-cluster theory -- 12.1 Spin summations and computational considerations -- 12.2 Coupled-cluster theory with an arbitrary single-determinant reference function -- 12.3 Generalized many-body perturbation theory -- 12.4 Brueckner orbitals and alternative treatments of ˆ T1 -- 12.5 Monitoring multiplicities in open-shell coupled-cluster calculations -- 12.6 The A and B response matrices from the viewpoint of CCS -- 12.7 Noniterative approximations based on the CC energy functional -- 12.8 The nature of the solutions of CC equations -- 13 The equation-of-motion coupled-cluster method for excited, ionized and electron-attached states -- 13.1 Introduction -- 13.2 The EOM-CC Ansatz -- 13.3 Diagrammatic treatment of the EE-EOM-CC equations -- 13.4 EOM-CC treatment of ionization and electron attachment -- 13.5 EOM-CC treatment of higher-order properties -- 13.6 EOM-CC treatment of frequency-dependent properties -- 14 Multireference coupled-cluster methods -- 14.1 Introduction -- 14.2 Hilbert-space state-universal MRCC -- 14.3 Hilbert-space state-specific MRCC -- 14.4 Fock-space valence-universal MRCC -- 14.4.1 The Fock-space approach -- 14.4.2 The valence-universal wave operator -- 14.4.3 The Fock-space Bloch equations. , 14.4.4 Relationship to EOM-CC -- 14.5 Intermediate-Hamiltonian Fock-space MRCC -- References -- Author index.