Preface ...................................................... xvii
CHAPTER 1 INTRODUCTION TO QUANTUM MECHANICS
1.1 What is Quantum Mechanics? ................................. 2
1.1.1 A Brief Early History of Quantum Mechanics .......... 2
1.1.2 Energy Quantization ................................. 2
1.1.3 Waves, Light, and Blackbody Radiation ............... 6
1.1.4 Wave-Particle Duality ............................... 8
1.1.5 Angular Momentum Quantization ...................... 12
1.1.6 Tunneling .......................................... 13
1.1.7 Photoelectric Effect ............................... 15
1.2 Nanotechnology and Information Technology ................. 17
1.2.1 STM and AFM Microscopies ........................... 18
1.2.2 Molecular Electronics .............................. 18
1.2.3 Quantum Dots, Wires and Wells, and Nanotubes ....... 19
1.2.4 Bio-Nanotechnology ................................. 21
1.2.5 Information Technology ............................. 21
1.3 A First Taste of Quantum Mechanics ........................ 21
1.3.1 Quantum States and Probability Distributions .......
1.3.2 Observable Operators ............................... 24
1.3.3 Quantum Entanglement ............................... 26
1.3.4 The Postulates of Quantum Mechanics ................ 28
1.3.5 Time-Dependent and -Independent Schrцdinger
Equations ..........................................
1.3.6 Momentum, Energy, and Angular Momentum ............. 32
1.3.7 Dirac Delta Functions .............................. 36
1.3.8 Position and Momentum States, |x› and |p› .......... 38
1.3.9 Ehrenfest's Theorem ................................ 39
1.3.10 One-Dimensional Wave Equations ..................... 41
1.3.11 Particle-in-a-Box and Piecewise-Constant
Potentials .........................................
1.3.12 The Delta Function Potential ....................... 52
1.3.13 Wave Packets ....................................... 53
1.3.14 The Linear Potential and Quantum Tunneling ......... 54
1.3.15 The Harmonic Oscillator ............................ 55
CHAPTER 2 THE FORMALISM OF QUANTUM MECHANICS .................. 61
2.1 Hilbert Space and Dirac Notation .......................... 61
2.1.1 Position and Momentum Representations .............. 63
2.1.2 Basis-State Expansions ............................. 64
2.2 Hermitian and Anti-Hermitian Operators .................... 66
2.2.1 Compatible Operators and Degeneracy ................ 66
2.3 The Uncertainty Principle ................................. 67
2.4 The Measurement Problem ................................... 69
2.5 Mixed States: Density Matrix Formulation .................. 70
2.5.1 Many-Particle Systems: Correlation Functions ....... 75
2.5.2 Purity and von Neumann Entropy ..................... 77
2.5.3 Distance Between States ............................ 77
2.5.4 The Measurement Problem Revisited .................. 78
2.6 The Wigner Representation ................................. 79
2.7 Schrцdinger and Heisenberg Representations ................ 85
2.7.1 Interaction Representation ......................... 87
2.7.2 Harmonic Oscillator Raising-Lowering Operators ..... 88
2.7.3 Coherent States and Squeezed States ................ 92
2.8 The Correspondence Principle and the Classical Limit ...... 97
2.9 Symmetry and Conservation Laws in Quantum Mechanics ....... 98
2.9.1 Exchange Symmetry .................................. 98
2.9.2 Inversion Symmetry ................................ 100
2.9.3 Time-Reversal Symmetry ............................ 100
2.9.4 Additional Generators of Galilean
Transformations ................................... 102
CHAPTER 3 ANGULAR MOMENTUM AND SPHERICAL SYMMETRY ............ 105
3.1 Angular Momentum in Quantum Mechanics .................... 105
3.1.1 Angular Momentum Raising and Lowering Operators ... 107
3.1.2 Electron Spin: j = 1/2 ............................ 110
3.1.3 Angular Momentum in Spherical Coordinates ......... 111
3.1.4 Spherical Harmonics ............................... 112
3.2 Spherically Symmetric Systems ............................ 116
3.2.1 Angular Momentum Decomposition of Plane Waves .....
3.2.2 Spherical Quantum Dot ............................. 120
3.2.3 The 3D Harmonic Oscillator ........................ 121
3.2.4 The Morse Oscillator .............................. 122
3.2.5 van der Waals and Lennard-Jones Potentials ........ 123
3.2.6 The Hydrogen Atom ................................. 124
3.3 Rotations and Angular Momentum ........................... 132
3.3.1 Euler angles α, β, γ and the Rotation Matrix ...... 133
3.3.2 Rotation and D Functions .......................... 134
3.3.3 Rigid-Rotor Eigenfunctions ........................ 137
3.4 Addition (Coupling) of Angular Momenta ................... 139
3.4.1 Clebsch-Gordan Coefficients and 3j Symbols ........ 140
3.4.2 Clebsch-Gordan Series ............................. 143
3.5 Tensor Operators ......................................... 144
3.5.1 Irreducible Representations of the Density
Matrix ............................................ 146
3.5.2 Vector Fields ..................................... 148
3.5.3 Spinor Fields ..................................... 149
3.5.4 Multipole Expansions .............................. 150
3.6 Symmetry Considerations .................................. 151
3.6.1 Selection Rules ................................... 152
3.6.2 Inversion Symmetry ................................ 152
3.6.3 Time-Reversal Symmetry ............................ 153
3.6.4 Wigner-Eckart Theorem ............................. 153
3.6.5 6j and Higher Coefficients ........................ 156
CHAPTER 4 SPIN ............................................... 159
4.1 Spin Angular Momentum .................................... 159
4.2 Spinors .................................................. 160
4.2.1 Pauli Matrices .................................... 161
4.2.2 Rotation of Spinors ............................... 164
4.2.3 Spin-Orbitals ..................................... 165
4.3 Electron in a Magnetic Field ............................. 166
4.3.1 Charged Particle in a Magnetic Field: Orbital
Effects ........................................... 169
4.4 Time-Reversal Properties of Spinors ...................... 172
4.5 Spin-Orbit Interaction in Atoms .......................... 175
4.6 Hyperfine Interaction .................................... 178
4.6.1 Electric Quadrupole Hyperfine Interaction ......... 181
4.6.2 Zeeman Splitting of Hyperfine States .............. 182
4.7 Spin-Dipolar Interactions ................................ 183
4.8 Introduction to Magnetic Resonance ....................... 185
4.8.1 The Rotating-Wave Approximation ................... 187
4.8.2 Spin Relaxation and the Bloch Equation ............ 188
4.8.3 Nuclear Spin Hamiltonian .......................... 189
4.8.4 Chemical Shifts ................................... 189
4.8.5 Fourier Transform NMR ............................. 190
CHAPTER 5 QUANTUM INFORMATION ................................ 193
5.1 Classical Computation and Classical Information .......... 194
5.1.1 Information and Entropy ........................... 194
5.1.2 Shannon Entropy ................................... 196
5.1.3 Data Compression .................................. 198
5.1.4 Classical Computers and Gates ..................... 200
5.1.5 Classical Cryptography ............................ 202
5.1.6 Computational Complexity .......................... 204
5.2 Quantum Information ...................................... 205
5.2.1 Qubits ............................................ 205
5.2.2 Quantum Entanglement and Bell States .............. 207
5.2.3 Quantum Gates ..................................... 213
5.2.4 No-Cloning Theorem ................................ 218
5.2.5 Dense Coding ...................................... 219
5.2.6 Data Compression of Quantum Information ........... 220
5.2.7 Quantum Teleportation ............................. 221
5.2.8 Quantum Cryptography .............................. 222
5.2.9 Quantum Circuits .................................. 223
5.2.10 Quantum Computing Despite Measurement ............. 224
5.3 Quantum Computing Algorithms ............................. 224
5.3.1 Deutsch and Deutsch-Jozsa Algorithms .............. 225
5.3.2 The Grover Search Algorithm ....................... 228
5.3.3 Quantum Fourier Transform ......................... 232
5.3.4 Shor Factorization Algorithm ...................... 234
5.3.5 Quantum Simulation ................................ 239
5.4 Decoherence .............................................. 240
5.5 Quantum Error Correction ................................. 240
5.6 Experimental Implementations ............................. 243
5.6.1 Ion Traps ......................................... 243
5.6.2 Neutral Atoms in Optical Lattices ................. 245
5.6.3 Cavity Based Quantum Computing .................... 246
5.6.4 Nuclear Magnetic Resonance Systems ................ 247
5.6.5 All-Optical Quantum Computers ..................... 248
5.6.6 Solid-State Qubits ................................ 248
5.7 The EPR Paradox .......................................... 250
5.8 Bell's Inequalities ...................................... 251
5.8.1 Bell's Inequalities and the EPR Paradox ........... 252
5.8.2 Bell's Analysis using Hidden Variables ............ 253
5.8.3 General Aspects of Bell's Inequalities ............ 256
CHAPTER 6 QUANTUM DYNAMICS AND CORRELATIONS ................... 259
6.1 Two-Level Systems ........................................ 259
6.1.1 Two-Level Dynamics (Spin Dynamics) ................ 261
6.1.2 The Bloch Sphere Picture .......................... 262
6.1.3 Coupling to a Bath: Decoherence ................... 266
6.1.4 Periodically Driven Two-Level System .............. 268
6.1.5 Atoms in an Electromagnetic Field: Dispersion
and Absorption .................................... 273
6.1.6 Doppler Cooling of Atoms .......................... 275
6.1.7 Optical Trapping of Atoms ......................... 278
6.1.8 Two or More Correlated "Spins" .................... 279
6.1.9 The N-Two-Level System Bloch Sphere ............... 282
6.1.10 Ramsey Fringe Spectroscopy ........................ 285
6.2 Three-Level Systems ...................................... 287
6.2.1 Two or More Three-Level Correlated Systems ........ 288
6.3 Classification of Correlation and Entanglement ........... 290
6.3.1 Entanglement Witness Operators .................... 292
6.4 Three-Level System Dynamics .............................. 293
6.5 Continuous-Variable Systems .............................. 295
6.6 Wave Packet Dynamics ..................................... 297
6.7 Time-Dependent Hamiltonians .............................. 299
6.8 Quantum Optimal Control Theory ........................... 300
CHAPTER 7 APPROXIMATION METHODS .............................. 303
7.1 Basis-State Expansions ................................... 303
7.1.1 Time-Dependent Basis Set Expansions ............... 303
7.2 Semiclassical Approximations ............................. 304
7.2.1 The WKB Approximation ............................. 304
7.2.2 Semiclassical Treatment of Dynamics ............... 309
7.2.3 Semiclassical Hamilton-Jacobi Expansion ........... 309
7.3 Perturbation Theory ...................................... 309
7.3.1 Nondegenerate Perturbation Theory ................. 310
7.3.2 Degenerate Perturbation Theory .................... 313
7.3.3 Time-Dependent Perturbation Theory ................ 315
7.4 Dynamics in an Electromagnetic Field ..................... 321
7.4.1 Spontaneous and Stimulated Emission of Radiation .. 323
7.4.2 Electric Dipole and Multipole Radiation ........... 324
7.4.3 Thomson, Rayleigh, Raman, and Brillouin
Transitions ....................................... 326
7.4.4 Decay Width ....................................... 329
7.4.5 Doppler Shift ..................................... 330
7.5 Exponential and Nonexponential Decay ..................... 331
7.6 The Variational Method ................................... 331
7.7 The Sudden Approximation ................................. 333
7.8 The Adiabatic Approximation .............................. 335
7.8.1 Chirped Pulse Adiabatic Passage ................... 336
7.8.2 Stimulated Raman Adiabatic Passage ................ 338
7.8.3 The Landau-Zener Problem .......................... 339
7.8.4 Generalized Displacements and Forces .............. 343
7.8.5 Berry Phase ....................................... 344
7.9 Linear Response Theory ................................... 349
7.9.1 Susceptibilities .................................. 350
7.9.2 Kubo Formulas ..................................... 352
7.9.3 Onsager Reciprocal Relations ...................... 360
7.9.4 Fluctuation-Dissipation Theorem ................... 362
CHAPTER 8 IDENTICAL PARTICLES ................................ 367
8.1 Permutation Symmetry ..................................... 367
8.1.1 The Symmetric Group Sn ............................ 369
8.1.2 Young Tableaux .................................... 370
8.2 Exchange Symmetry ........................................ 371
8.2.1 Symmetrization Postulate .......................... 372
8.3 Permanents and Slater Determinants ....................... 374
8.4 Simple Two- and Three-Electron States .................... 375
8.5 Exchange Symmetry for Two Two-Level Systems .............. 377
8.6 Many-Particle Exchange Symmetry .......................... 378
CHAPTER 9 ELECTRONIC PROPERTIES OF SOLIDS .................... 381
9.1 The Free Electron Gas .................................... 381
9.1.1 Density of States in 2D and 1D Systems ............ 386
9.1.2 Fermi-Dirac Distribution .......................... 388
9.2 Elementary Theories of Conductivity ...................... 391
9.2.1 Drude Theory of Conductivity ...................... 392
9.2.2 Thermal Conductivity of Metals .................... 397
9.2.3 Sommerfeld Theory of Transport in Metals .......... 399
9.3 Crystal Structure ........................................ 400
9.3.1 Bravais Lattices and Crystal Systems .............. 401
9.3.2 The Reciprocal Lattice ............................ 405
9.3.3 Quasicrystals ..................................... 412
9.4 Electrons in a Periodic Potential ........................ 414
9.4.1 From Atomic Orbits to Band Structure .............. 414
9.4.2 Band Structure and Electron Transport ............. 415
9.4.3 Periodic Potential and Band Formation ............. 416
9.4.4 Bloch Wave Functions and Energy Bands ............. 418
9.4.5 Schrödinger Equation in Reciprocal Lattice Space .. 422
9.4.6 Sinusoidal Potential: Mathieu Functions ........... 423
9.4.7 Tight-Binding Model ............................... 427
9.4.8 Wannier Functions ................................. 429
9.4.9 Electric Field Effects ............................ 430
9.5 Magnetic Field Effects ................................... 434
9.5.1 Electron in a Magnetic Field ...................... 435
9.5.2 Aharonov-Bohm Effect .............................. 436
9.5.3 The Hall Effect and Magnetoresistance ............. 443
9.5.4 Landau Quantization ............................... 445
9.5.5 2D Electron Gas in a Perpendicular Magnetic
Field ............................................. 448
9.5.6 Electron Subject to Periodic Potential and
Magnetic Field .................................... 454
9.5.7 de Haas-van Alphen and Shubnikov-de Haas Effects .. 459
9.5.8 The Quantum Hall Effect ........................... 462
9.5.9 Paramagnetism and Diamagnetism .................... 468
9.5.10 Magnetic Order .................................... 475
9.6 Semiconductors ........................................... 481
9.6.1 Semiconductor Band Structure ...................... 482
9.6.2 Charge Carrier Density ............................ 484
9.6.3 Extrinsic Semiconductors .......................... 486
9.6.4 Inhomogeneous Semiconductors: p-n Junctions ....... 490
9.6.5 Excitons .......................................... 500
9.6.6 Spin-Orbit Coupling in Solids ..................... 504
9.6.7 к • p Perturbation Theory ......................... 506
9.6.8 Spin Hall Effect .................................. 510
9.6.9 Photon Induced Processes in Semiconductors ........ 512
9.7 Spintronics .............................................. 515
9.7.1 Tools for Manipulating Spins ...................... 516
9.7.2 Tunneling Magnetoresistance ....................... 520
9.7.3 Spintronic Devices ................................ 523
9.8 Low-Energy Excitations ................................... 526
9.8.1 Phonons ........................................... 527
9.8.2 Plasmons .......................................... 531
9.8.3 Magnons ........................................... 535
9.8.4 Polarons .......................................... 538
9.8.5 Polaritons ........................................ 541
9.9 Insulators ............................................... 541
9.9.1 Defining Insulators ............................... 542
9.9.2 Classification of Insulators ...................... 543
CHAPTER 10 ELECTRONIC STRUCTURE OF MULTIELECTRON SYSTEMS ...... 545
10.1 The Multielectron System Hamiltonian ..................... 545
10.2 Slater and Gaussian Type Atomic Orbitals ................. 546
10.3 Term Symbols for Atoms ................................... 547
10.4 Two-Electron Systems ..................................... 547
10.4.1 The Helium Atom ................................... 548
10.4.2 The Hartree Method: Helium ........................ 548
10.5 Hartree Approximation for Multielectron Systems .......... 551
10.6 The Hartree-Fock Method .................................. 552
10.6.1 Hartree-Fock for Helium ........................... 557
10.7 Koopmans' Theorem ........................................ 558
10.8 Atomic Radii ............................................. 560
10.9 Multielectron Fine Structure: Hund's Rules ............... 560
10.10 Electronic Structure of Molecules ....................... 562
10.10.1 H2+: Molecular Orbitals .......................... 563
10.10.2 The Hydrogen Molecule ............................ 565
10.10.3 The Hiickel Approximation ........................ 569
10.11 Hartree-Fock for Metals ................................. 572
10.12 Electron Correlation .................................... 574
10.12.1 Configuration Interaction ........................ 574
10.12.2 Moller-Plesset Many-Body Perturbation Theory ..... 576
10.12.3 Coupled Cluster Method ........................... 577
CHAPTER 11 MOLECULES .......................................... 579
11.1 Molecular Symmetries ..................................... 579
11.1.1 Molecular Orbitals and Group Theory ............... 580
11.1.2 Character Tables and Mulliken Symbols ............. 581
11.2 Diatomic Electronic States ............................... 582
11.2.1 Interatomic Potentials at Large Internuclear
Distances ......................................... 585
11.2.2 Hund's Coupling ................................... 586
11.2.3 Hyperfine Interactions in Diatomic Molecules ...... 589
11.3 The Born-Oppenheimer Approximation ....................... 589
11.3.1 Potential Energy Crossings and Pseudocrossings .... 590
11.3.2 Born-Oppenheimer Nuclear Derivative Coupling ...... 591
11.3.3 The Hellman-Feynman Theorem ....................... 592
11.4 Rotational and Vibrational Structure ..................... 593
11.5 Vibrational Modes and Symmetry ........................... 595
11.6 Selection Rules for Optical Transitions .................. 596
11.6.1 Selection Rules for Diatomic Molecules ........... 597
11.7 The Franck-Condon Principle .............................. 600
11.7.1 Bound-Free Matrix Elements ........................ 602
CHAPTER 12 SCATTERING THEORY .................................. 605
12.1 Classical Scattering Theory .............................. 605
12.2 Quantum Scattering ....................................... 609
12.2.1 Time-Dependent and Stationary Approaches ..........
12.2.2 Preparation of the Initial State .................. 611
12.2.3 Time-Dependent Formulation ........................ 612
12.3 Stationary Scattering Theory ............................. 616
12.3.1 Cross-Sections .................................... 617
12.3.2 Two-Body Collisions ............................... 618
12.3.3 From Wave Functions to Cross-Sections ............. 622
12.3.4 Green's Function .................................. 623
12.4 Aspects of Formal Scattering Theory ...................... 627
12.4.1 The Transition Operator (The T Matrix) ............ 628
12.4.2 The 5 Matrix and Möller Operators ................. 633
12.5 Central Potentials ....................................... 635
12.5.1 Central Potentials and Spin ....................... 635
12.5.2 Axial Symmetry .................................... 635
12.5.3 Partial Wave Analysis ............................. 636
12.5.4 Phase Shift Analysis .............................. 647
12.5.5 Scattering from a Coulomb Potential ............... 659
12.5.6 Scattering of Two Identical Particles ............. 664
12.6 Resonance Scattering ..................................... 666
12.6.1 Influence of Bound States ......................... 666
12.6.2 Resonance Cross-Sections .......................... 668
12.6.3 Feshbach Resonance ................................ 669
12.6.4 Fano Resonance .................................... 677
12.7 Approximation Methods .................................... 682
12.7.1 Bora Approximation ................................ 683
12.7.2 WKB Approximation ................................. 684
12.7.3 The Variational Principle ......................... 688
12.7.4 Eikonal Approximation ............................. 690
12.8 Particles with Internal Degrees of Freedom ............... 693
12.8.1 Spin .............................................. 693
12.8.2 Composite Particles ............................... 693
12.8.3 Channels .......................................... 693
12.8.4 Asymptotic States and Cross-Sections .............. 695
12.8.5 Möller Operators .................................. 696
12.8.6 The Multichannel S Matrix ......................... 699
12.8.7 Scattering from Two Potentials .................... 703
12.8.8 Scattering of Particles with Spin ................. 704
12.8.9 Inelastic Scattering and Scattering Reactions ..... 715
12.8.10 Scattering from a Collection of Identical
Particles ......................................... 722
12.9 Scattering in Low-Dimensional Systems .................... 724
12.9.1 Scattering in Two Dimensions ...................... 724
12.9.2 ID Scattering: S Matrix ........................... 731
12.9.3 ID Scattering: Anderson Localization .............. 737
12.9.4 Scattering in Quasi-One-Dimensional Systems ....... 741
CHAPTER 13 LOW-DIMENSIONAL QUANTUM SYSTEMS .................... 749
13.1 Mesoscopic Systems ....................................... 750
13.1.1 Low-Dimensional Nanostructures .................... 752
13.1.2 Quantum Wells ..................................... 752
13.1.3 Quantum Wires ..................................... 753
13.1.4 Quantum Dots ...................................... 754
13.1.5 Heterojunctions and Superlattices ................. 754
13.1.6 Quantum Point Contacts ............................ 755
13.2 The Landauer Conductance Formula ......................... 755
13.2.1 Aharonov-Bohm Interferometer ...................... 757
13.2.2 Multiport Landauer Formula ........................ 760
13.2.3 Conductance of Quantum Point Contacts ............. 762
13.3 Properties of Quantum Dots ............................... 764
13.3.1 Equilibrium Properties of Quantum Dots ............ 765
13.3.2 Charge Transport through Quantum Dots ............. 770
13.4 Disorder in Mesoscopic Systems ........................... 773
13.4.1 Disorder in Quantum Dots .......................... 773
13.4.2 Disordered Systems and Random Matrices ............ 774
13.4.3 Application of Random Matrix Theory to
Disordered Mesoscopic Systems ..................... 777
13.5 Kondo Effect in Quantum Dots ............................. 783
13.5.1 The Dot Kondo Hamiltonian ......................... 784
13.5.2 The Dot Kondo Temperature ......................... 785
13.5.3 Abrikosov-Suhl Resonance and Kondo Conductance .... 785
13.6 Graphene ................................................. 787
13.6.1 Carbon Nanotubes .................................. 788
13.6.2 Lattice Structure and Dirac Cones ................. 790
13.6.3 Dirac Equation and its Relevance to Graphene ...... 792
13.6.4 Tight-Binding Model for Graphene .................. 797
13.6.5 Continuum Theory .................................. 800
13.6.6 Landau Levels in Graphene ......................... 804
13.6.7 Potential Scattering in Graphene .................. 806
13.7 Inventory of Recently Discovered Low-Dimensional
Phenomena ................................................ 810
13.7.1 Persistent Currents ............................... 811
13.7.2 Weak Localization ................................. 811
13.7.3 Shot Noise ........................................ 813
13.7.4 Strongly Correlated Low-Dimensional Systems ....... 815
13.7.5 Wigner Crystals ................................... 815
13.7.6 The Fractional Quantum Hall Effect ................ 816
13.7.7 High-Temperature Superconductivity and 2D
Magnetism ......................................... 819
13.7.8 ID Correlated Electron Systems .................... 820
13.7.9 ID Spin Chains .................................... 821
13.7.10 Quantum Spin Hall Effect ......................... 822
CHAPTER 14 MANY-BODY THEORY ................................... 825
14.1 Second Quantization ...................................... 825
14.1.1 Interacting Identical Particles: A Generic
Many-Body Problem ................................. 826
14.1.2 A Basis for Many-Body Wave Functions .............. 827
14.1.3 Mapping onto Fock Space ........................... 830
14.1.4 Creation and Annihilation Operators ............... 832
14.1.5 The Hamiltonian in Fock Space ..................... 837
14.1.6 Field Operators ................................... 843
14.1.7 Electromagnetic Field Quantization: Photons ....... 850
14.1.8 Quantization of Lattice Vibrations: Phonons ....... 853
14.1.9 Systems with Two Kinds of Particles ............... 856
14.2 Statistical Mechanics in Second Quantization ............. 857
14.3 The Electron Gas ......................................... 860
14.3.1 Electrons in the Jellium Model ................... 861
14.4 Mean-Field Theory ........................................ 866
14.4.1 Mean-Field Equations in Second Quantization ...... 867
CHAPTER 15 DENSITY FUNCTIONAL THEORY .......................... 871
15.1 The Hohenberg-Kohn Theorems .............................. 872
15.2 The Thomas-Fermi Approximation ........................... 875
15.3 The Kohn-Sham Equations .................................. 878
15.3.1 Local Density Approximation ....................... 881
15.4 Spin DFT and Magnetic Systems ............................ 882
15.4.1 Spin DFT Local Density Approximation ............. 883
15.5 The Gap Problem in DFT ................................... 884
15.6 Time-Dependent DFT ....................................... 885
15.6.1 The Runge-Gross Theorem ........................... 886
15.6.2 Time-Dependent Kohn-Sham Equations ................ 888
15.6.3 Adiabatic LDA ..................................... 888
15.7 DFT Computer Packages .................................... 888
APPENDIX A LINEAR ALGEBRA ..................................... 891
A.l Vector Spaces ............................................ 891
A.1.1 Dirac Notation .................................... 892
A.1.2 Inner Product Spaces .............................. 893
A.2 Operators and Matrices ................................... 898
A.2.1 Outer Product ..................................... 899
A.2.2 Determinants and Permanents ....................... 900
A.2.3 Normal, Unitary, Hermitian-Conjugate and
Hermitian Operators ............................... 901
A.2.4 Trace and Projection .............................. 902
A.2.5 Antilinear and Antiunitary Operators .............. 903
APPENDIX В SOME ORDINARY DIFFERENTIAL EQUATIONS ............... 905
APPENDIX С VECTOR ANALYSIS .................................... 913
C.1 Scalar and Vector Products ............................... 913
C.2 Differential Operators ................................... 914
C.3 Divergence and Stokes Theorems ........................... 915
C.4 Curvilinear Coordinates .................................. 916
APPENDIX D FOURIER ANALYSIS ................................... 921
D.1 Fourier Series ........................................... 922
D.1.1 Fourier Series of Functions of a Discrete
Variable .......................................... 924
D.2 Fourier Integrals ........................................ 925
D.3 Fourier Series and Integrals in Three-Space Dimensions ... 927
D.3.1 3D Fourier Integrals ............................... 927
D.4 Fourier Integrals of Time-Dependent Functions ............ 928
D.5 Convolution .............................................. 928
D.6 Fourier Expansion of Operators ........................... 929
D.7 Fourier Transforms ....................................... 929
D.8 FT for Solving Differential and Integral Equations ....... 931
APENDIX E SYMMETRY AND GROUP THEORY ........................... 933
E.1 Group Theory Axioms ...................................... 933
E.2 Group Multiplication Tables .............................. 933
E.3 Examples of Groups ....................................... 934
E.3.1 Point Groups ....................................... 935
E.3.2 Space Groups ....................................... 936
E.3.3 Continuous Groups ................................. 936
E.4 Some Properties of Groups ................................ 936
E.5 Group Representations .................................... 937
E.5.1 Irreducible Representations ....................... 938
E.5.2 Group Orthogonality Theorem ....................... 939
E.5.3 Characters and Character Tables ................... 939
E.5.4 Constructing Irreducible Representations .......... 942
Bibliography .................................................. 943
Index ......................................................... 953
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