Schrödinger theory of electrons : complementary perspectives

목차 (Table of Contents)

• CONTENTS
• 1 Introduction = 1
• References = 16
• 2 Schrödinger Theory of Electrons : A Complementary Perspective = 17
• 2.1 Time-Dependent Schrödinger Theory = 19
• 2.2 Definitions of Quantal Sources = 21
• 2.2.1 Electron Density ρ(rt) = 21
• 2.2.2 Spinless Single-Particle Density Matrix γ (rr't) = 22
• 2.2.3 Pair-Correlation Density g(rr't), And Fermi-Coulomb Hole ρxc(rr't) = 23
• 2.2.4 Current Density j(rt) = 25
• 2.3 Definitions of ‘Classical’ Fields = 26
• 2.3.1 Electron-Interaction Field Eee(rt) = 26
• 2.3.2 Differential Density Field D(rt) = 27
• 2.3.3 Kinetic Field Z(rt) = 27
• 2.3.4 Current Density Field J (rt) = 28
• 2.4 Energy Components in Terms of Quantal Sources and Fields = 28
• 2.4.1 Electron-Interaction Potential Energy Eee(t) = 29
• 2.4.2 Kinetic Energy T (t) = 30
• 2.4.3 External Potential Energy Eext(t) = 31
• 2.5 The ‘Quantal Newtonian’ Second Law = 31
• 2.6 The Internal Field and Ehrenfest’s Theorem = 33
• 2.7 Integral Virial Theorem = 37
• 2.8 Time-Independent Schrödinger Theory : Ground and Bound Excited States = 39
• 2.8.1 The Quantal-Source and Field Perspective = 39
• 2.8.2 Energy Components in Terms of Quantal Sources and Fields 40
• 2.8.3 The ‘Quantal-Newtonian’ First Law = 41
• 2.8.4 Integral Virial and Ehrenfest’s Theorems = 42
• 2.9 Remarks on Quantum Fluid Dynamics and the ‘Quantal Newtonian’ Laws = 45
• References = 47
• 3 Generalization of the Schrödinger Theory of Electrons = 49
• 3.1 Generalization of the Stationary-State Schrödinger Equation = 54
• 3.1.1 ‘Quantal Newtonian’ First Law in an Electrostatic and Magnetostatic Field = 56
• 3.1.2 New Insights to the Stationary-State Schrödinger Equation = 59
• 3.2 Hartree-Fock and Hartree Theories in Terms of Quantal Sources and Fields = 63
• 3.2.1 Hartree-Fock Theory in Terms of Quantal Sources and Fields = 66
• 3.2.2 Derivation of the Hartree-Fock Theory Integro-Differential Equation = 71
• 3.2.3 The Slater-Bardeen Interpretation of Hartree-Fock Theory = 74
• 3.2.4 ‘Quantal Newtonian’ First Law in Hartree-Fock Theory = 76
• 3.2.5 Generalization of the Hartree-Fock Theory Equations = 77
• 3.2.6 Theorems in Hartree-Fock Theory = 79
• 3.2.7 Hartree Theory in Terms of Quantal Sources and Fields = 81
• 3.2.8 Derivation of the Hartree Theory Integro-Differential Equation = 85
• 3.2.9 ‘Quantal Newtonian’ First Law in Hartree Theory = 87
• 3.2.10 Generalization of the Hartree Theory Equations = 88
• 3.3 Generalization of the Time-Dependent Schrödinger Equation = 89
• References = 91
• 4 Schrödinger-Pauli Theory of Electrons : A Complementary Perspective = 93
• 4.1 The Classical Hamiltonian in an Electromagnetic Field = 95
• 4.2 Stationary-State Schrödinger Theory in an Electromagnetic Field = 98
• 4.2.1 Schrödinger Theory Hamiltonian = 99
• 4.2.2 Magnetic Field-Orbital Angular Momentum Interaction = 100
• 4.2.3 Schrödinger Theory in Terms of the Density and Physical Current Density = 101
• 4.2.4 Schrödinger Theory Hamiltonian in Terms of the Lorentz ‘Force’ Operator = 103
• 4.2.5 The Wave Function, a Functional of the Gauge Function = 104
• 4.3 Stationary-State Schrödinger-Pauli Theory in an Electromagnetic Field = 106
• 4.3.1 Schrödinger-Pauli Theory Hamiltonian and Equation = 107
• 4.3.2 Schrödinger-Pauli Theory in Terms of the Density and Physical Current Density = 108
• 4.4 Stationary-State Schrödinger-Pauli Theory in Terms of Quantal Sources and Fields = 111
• 4.4.1 The ‘Quantal Newtonian’ First Law for an Electron with Spin = 112
• 4.4.2 Energy Components in Terms of Fields = 115
• 4.4.3 Physical and Mathematical Insights = 117
• 4.4.4 Generalization of the Schrödinger-Pauli Theory Equation = 119
• 4.5 Time-Dependent Schrödinger-Pauli Theory and Its Generalization : The ‘Quantal Newtonian’ Second Law for an Electron with Spin = 121
• References = 124
• 5 Elucidation of Complimentary Perspective to Schrödinger-Pauli Theory : Application to the 23S
• State of a Quantum Dot in a Magnetic Field = 127
• 5.1 Triplet 23 S State Wave Function = 129
• 5.2 Quantal Sources = 134
• 5.2.1 Electron Density ρ(r) = 134
• 5.2.2 Physical Current Density j(r) and Its Paramagnetic jp(r), Diamagnetic jd (r) and Magnetization jm(r)
• Components = 136
• 5.2.3 Pair-Correlation Density g(rr) and the Fermi-Coulomb Hole ρxc(rr) = 138
• 5.2.4 Single-Particle Density Matrix γ (rr) = 141
• 5.3 ‘Forces’, Fields, and Energies = 143
• 5.3.1 Electron-Interaction, Hartree, Pauli-Coulomb = 143
• 5.3.2 Kinetic = 146
• 5.3.3 Differential Density = 148
• 5.3.4 Lorentz, Internal Magnetic, and External Electrostatic = 148
• 5.4 Total Energy E and Ionization Potential I P = 153
• 5.5 Expectation Values of Single-Particle Operators = 154
• 5.6 Satisfaction of the ‘Quantal Newtonian’ First Law = 154
• 5.7 Self-Consistent Nature of the Schrödinger-Pauli Equation = 157
• References = 159
• 6 Quantal Density Functional Theory : A Local Effective Potential Theory Complement to Schrödinger Theory = 161
• 6.1 Stationary-State Quantal Density Functional Theory = 172
• 6.1.1 Quantal Sources = 172
• 6.1.2 ‘Classical’ Fields Experienced by Each Model Fermion = 175
• 6.1.3 The S System ‘Quantal Newtonian’ First Law = 180
• 6.1.4 Effective Field Feff(r) and Electron-Interaction Potential vee(r) = 181
• 6.1.5 Total Energy E in Terms of S System Properties = 183
• 6.1.6 Sum Rules Satisfied by the Effective Field Feff(r) = 185
• 6.1.7 Proof that Nonuniqueness of Effective Potential Energy is Solely Due to Correlation-Kinetic Effects = 186
• 6.1.8 Physical Interpretation of Highest Occupied Eigenvalue Em = 187
• 6.2 Application of Q-DFT to the Ground and First Excited Singlet State of a Quantum Dot in a Magnetic Field = 188
• 6.2.1 Interacting Electronic System : The Quantum Dot = 189
• 6.2.2 Noninteracting Model Fermion System = 192
• 6.2.3 Quantal Sources = 193
• 6.2.4 Fields, Potentials, Energies, and Eigenvalues = 199
• 6.2.5 Quantal Density Functional Theory of the Density Amplitude = 207
• 6.3 Time-Dependent Quantal Density Functional Theory = 208
• 6.3.1 The S System ‘Quantal Newtonian’ Second Law = 209
• 6.3.2 Effective Field Feff(y) and Electron-Interaction Potential vee(y) = 209
• References = 212
• 7 Modern Density Functional Theory = 215
• 7.1 Paths to the Hamiltonian = 216
• 7.2 The First Hohenberg-Kohn Theorem = 222
• 7.2.1 The Gunnarsson-Lundqvist Theorem for Excited States = 225
• 7.2.2 The Inverse Maps C−1 and D−1 = 226
• 7.2.3 Generalization of the First Hohenberg-Kohn Theorem via Density Preserving Unitary Transformations = 228
• 7.2.4 Corollary to the First Hohenberg-Kohn Theorem = 230
• 7.3 The Second Hohenberg-Kohn Theorem = 236
• 7.3.1 Physical Interpretation of Lagrange Multiplier μ = 237
• 7.3.2 The Primacy of Electron Number N = 238
• 7.3.3 The Percus-Levy-Lieb Constrained-Search Proof = 239
• 7.3.4 Comment on the Constrained-Search Definition of the Functional FHK[ρ] = 240
• 7.4 Kohn-Sham Density Functional Theory = 240
• 7.5 Physical Interpretation of Kohn-Sham Theory = 246
• 7.5.1 Electron Correlations in Kohn-Sham ‘Exchange’ and ‘Correlation’ = 250
• 7.5.2 Definitions of the Correlation Energy = 251
• 7.5.3 Electron Correlations in Approximate Kohn-Sham Theory = 252
• 7.6 The Hohenberg-Kohn Theorems in a Uniform Magnetic Field = 253
• 7.6.1 The First Hohenberg-Kohn Theorem : Case 1 : Spinless Electrons = 254
• 7.6.2 The First Hohenberg-Kohn Theorem : Case 2 : Electrons With Spin = 260
• 7.6.3 The Second Hohenberg-Kohn Theorem = 263
• 7.6.4 The Percus-Levy-Lieb Constrained-Search Proof = 264
• 7.6.5 Remarks on Basic Variables in a Magnetic Field = 265
• 7.7 Time-Dependent Density Functional Theory = 267
• 7.7.1 The First Runge-Gross Theorem = 268
• 7.7.2 Generalization of the First Runge-Gross Theorem Via Density Preserving Unitary Transformation = 269
• 7.7.3 Corollary to the First Runge-Gross Theorem = 271
• 7.7.4 The van Leeuwen Theorem = 273
• 7.7.5 Physical Interpretation of Time-Dependent Kohn-Sham Theory = 276
• References = 277
• 8 Wave Function Properties = 281
• 8.1 Coalescence Constraints in Dimensions D ≥ 2 = 284
• 8.1.1 Differential Form of Electron-Nucleus Coalescence Condition in Terms of the D ensity = 288
• 8.1.2 Coalescence Constraints for the Pair-Correlation Function = 289
• 8.1.3 Significance of Electron-Nucleus Coalescence Constraint to Local Effective Potential Theories = 290
• 8.2 Asymptotic Structure of the Wave Function in the Classically Forbidden Region = 294
• 8.2.1 Asymptotic Structure of the Density and Sum Rule = 296
• 8.2.2 Significance of Asymptotic Structure of Wave Function and Density to Local Effective Potential
• Theories = 297
• 8.3 A New Symmetry for 2-Electron Systems in an Electromagnetic Field = 299
• 8.3.1 Permutation Operation and the Pauli Principle = 302
• 8.3.2 New Symmetry Operation and Wave Function Identity = 305
• 8.3.3 Parity of Singlet and Triplet State Wave Functions = 309
• 8.3.4 Parity About All Points of Electron-Electron Coalescence = 312
• 8.3.5 Proof of Satisfaction of the Wave Function Identity By the Exact Wave Function = 313
• References = 316
• 9 Wave Functions for Harmonically Bound Electrons in an Electromagnetic Field = 319
• 9.1 Wave Functions for the 2D 2-Electron Harmonically Bound ‘Artificial Atom’ in a Magnetic Field = 322
• 9.1.1 Decoupling of the Hamiltonian = 323
• 9.1.2 The Relative Coordinate Wave Function Component φ(s) = 326
• 9.1.3 The Center of Mass Coordinate Wave Function Component Φ(R) = 336
• 9.1.4 The General Wave Function Ψ (x1x2) and Total Energy E = 336
• 9.1.5 Wave Function for a Ground and First Excited Triplet State = 338
• 9.2 The Generalized Kohn Theorem Wave Function = 342
• 9.2.1 Decoupling of the Hamiltonian = 345
• 9.2.2 In-plane Center-of-Mass Motion = 346
• 9.2.3 Evolution of the Eigenstates of the In-plane Motion = 352
• 9.2.4 Total Wave Function and Classical Equation of Motion = 354
• 9.2.5 The Relative Coordinate Hamiltonian = 356
• References = 361
• 10 Epilogue 363
• Appendix A = 369
• Appendix B = 381
• Appendix C = 389
• Appendix D = 395
• Index = 403