Introduction To Quantum Mechanics With Applications To Chemistry

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Introduction To Quantum Mechanics With Applications To Chemistry......Page 1 Title Page......Page 2 Copyright......Page 3 Preface, July 1935......Page 4 Contents......Page 6 1. Survey of Classical Mechanics......Page 14 1.1 Newton's Equations of Motion in the Lagrangian Form......Page 15 1.1a The Three-Dimensional Isotropic Harmonic Oscillator......Page 17 1.1b Generalized Coordinates......Page 19 1.1c The Invariance of the Equations of Motion in the Lagrangian Form......Page 20 1.1d An Example: The Isotropic Harmonic Oscillator in Polar Coordinates......Page 22 1.1e The Conservation of Angular Momentum......Page 24 1.2a Generalized Momenta......Page 27 1.2c The Hamiltonian Function and the Energy......Page 29 1.2d A General Example......Page 30 1.3 The Emission and Absorption of Radiation......Page 34 1.4 Summary of Chapter I......Page 36 2.5 The Origin of the Old Quantum Theory......Page 38 2.5a The Postulates of Bohr......Page 39 2.5b The Wilson-Sommerfeld Rules of Quantization......Page 41 2.5c Selection Rules. The Correspondence Principle......Page 42 2.6a The Harmonic Oscillator. Degenerate States......Page 43 2.6b The Rigid Rotator......Page 44 2.6c The Oscillating and Rotating Diatomic Molecule......Page 45 2.6d The Particle in a Box......Page 46 2.6e Diffraction by a Crystal Lattice......Page 47 2.7a Solution of the Equations of Motion......Page 49 2.7b Application of the Quantum Rules. The Energy Levels......Page 52 2.7c Description of the Orbits......Page 56 2.7d Spatial Quantization......Page 58 2.8 The Decline of the Old Quantum Theory......Page 60 3.9 The Schrödinger Wave Equation......Page 63 3.9a The Wave Equation Including the Time......Page 66 3.9b The Amplitude Equation......Page 69 3.9c Wave Functions. Discrete and Continuous Sets of Characteristic Energy Values......Page 71 3.10a psi^*x, t psix, t as a Probability Distribution Function......Page 76 3.10b Stationary States......Page 77 3.10c Further Physical Interpretation. Average Values of Dynamical Quantities......Page 78 3.11a Solution of the Wave Equation......Page 80 3.11b The Wave Functions for the Harmonic Oscillator and Their Physical Interpretation......Page 86 3.11c Mathematical Properties of the Harmonic Oscillator Wave Functions......Page 90 4.12 The Wave Equation for a System of Point Particles......Page 97 4.12a The Wave Equation Including the Time......Page 98 4.12b The Amplitude Equation......Page 99 4.12d The Physical Interpretation of the Wave Functions......Page 101 4.13 The Free Particle......Page 103 4.14 The Particle in a Box......Page 108 4.15 The Three-Dimensional Harmonic Oscillator in Cartesian Coordinates......Page 113 4.16 Curvilinear Coordinates......Page 116 4.17 The Three-Dimensional Harmonic Oscillator in Cylindrical Coordinates......Page 118 5. The Hydrogen Atom......Page 125 5.18a The Separation of the Wave Equation. The Translational Motion......Page 126 5.18b The Solution of the phi Equation......Page 130 5.18c The Solution of the thetasym Equation......Page 131 5.18d The Solution of the r Equation......Page 134 5.18e The Energy Levels......Page 137 5.19 Legendre Functions and Surface Harmonics......Page 138 5.19a The Legendre Functions or Legendre Polynomials......Page 139 5.19b The Associated Legendre Functions......Page 140 5.20a The Laguerre Polynomials......Page 142 5.20b The Associated Laguerre Polynomials and Functions......Page 144 5.21a Hydrogenlike Wave Functions......Page 145 5.21b The Normal State of the Hydrogen Atom......Page 152 5.21c Discussion of the Hydrogenlike Radial Wave Functions......Page 155 5.21d Discussion of the Dependence of the Wave Functions on the Angles thetasym and phi......Page 159 6.22 Expansions in Series of Orthogonal Functions......Page 164 6.23 First-Order Perturbation Theory for a Non-Degenerate Level......Page 169 6.23a A Simple Example: The Perturbed Harmonic Oscillator......Page 173 6.23b An Example: The Normal Helium Atom......Page 175 6.24 First-Order Perturbation Theory for a Degenerate Level......Page 178 6.24a An Example: Application of a Perturbation to a Hydrogen Atom......Page 185 6.25 Second-Order Perturbation Theory......Page 189 6.25a An Example: The Stark Effect of the Plane Rotator......Page 190 7.26a The Variational Integral and its Properties......Page 193 7.26b An Example: The Normal State of the Helium Atom......Page 197 7.26d Linear Variation Functions......Page 199 7.26e A More General Variation Method......Page 202 7.27a A Generalized Perturbation Theory......Page 204 7.27b The Wentzel-Kramers-Brillouin Method......Page 211 7.27c Numerical Integration......Page 214 7.27d Approximation by the Use of Difference Equations......Page 215 7.27e An Approximate Second-Order Perturbation Treatment......Page 217 8.28 The Spinning Electron......Page 220 8.29a The Configurations 1s2s and 1s2p......Page 223 8.29b The Consideration of Electron Spin. The Pauli Exclusion Principle......Page 227 8.29c The Accurate Treatment of the Normal Helium Atom......Page 234 8.29d Excited States of the Helium Atom......Page 238 8.29e The Polarizability of the Normal Helium Atom......Page 239 9.30a Exchange Degeneracy......Page 243 9.30b Spatial Degeneracy......Page 246 9.30c Factorization and Solution of the Secular Equation......Page 248 9.30d Evaluation of Integrals......Page 252 9.30e Empirical Evaluation of Integrals. Applications......Page 257 9.31 Variation Treatments for Simple Atoms......Page 259 9.31a The Lithium Atom and Three-Electron Ions......Page 260 9.31b Variation Treatments of other Atoms......Page 262 9.32a Principle of the Method......Page 263 9.32b Relation of the Self-Consistent Field Method to the Variation Principle......Page 265 9.32c Results of the Self-Consistent Field Method......Page 267 9.33a Semi-Empirical Sets of Screening Constants......Page 269 9.33b The Thomas-Fermi Statistical Atom......Page 270 10.34 The Separation of Electronic and Nuclear Motion......Page 272 10.35 The Rotation and Vibration of Diatomic Molecules......Page 276 10.35a The Separation of Variables and Solution of the Angular Equations......Page 277 10.35b The Nature of the Electronic Energy Function......Page 279 10.35c A Simple Potential Function for Diatomic Molecules......Page 280 10.35d A More Accurate Treatment. The Morse Function......Page 284 10.36a The Rotation of Symmetrical-Top Molecules......Page 288 10.36b The Rotation of Unsymmetrical-Top Molecules......Page 293 10.37a Normal Coordinates in Classical Mechanics......Page 295 10.37b Normal Coordinates in Quantum Mechanics......Page 301 10.38 The Rotation of Molecules in Crystals......Page 303 11.39 The Treatment of a Time-Dependent Perturbation by the Method of Variation of Constants......Page 307 11.39a A Simple Example......Page 309 11.40a The Einstein Transition Probabilities......Page 312 11.40b The Calculation of the Einstein Transition Probabilities by Perturbation Theory......Page 315 11.40d Selection Rules and Intensities for Surface-Harmonic Wave Functions......Page 319 11.40e Selection Rules and Intensities for the Diatomic Molecule. The Franck-Condon Principle......Page 322 11.40f Selection Rules and Intensities for the Hydrogen Atom......Page 325 11.40g Even and Odd Electronic States and Their Selection Rules......Page 326 11.41 The Resonance Phenomenon......Page 327 11.41a Resonance in Classical Mechanics......Page 328 11.41b Resonance in Quantum Mechanics......Page 331 11.41c. A Further Discussion of Resonance......Page 335 12. The Structure of Simple Molecules......Page 339 12.42a A Very Simple Discussion......Page 340 12.42b Other Simple Variation Treatments......Page 344 12.42c The Separation and Solution of the Wave Equation......Page 346 12.43a The Treatment of Heitler and London......Page 353 12.43b Other Simple Variation Treatments......Page 358 12.43c The Treatment of James and Coolidge......Page 362 12.43d Comparison with Experiment......Page 364 12.43e Excited States of the Hydrogen Molecule......Page 366 12.43f Oscillation and Rotation of the Molecule. Ortho and Para Hydrogen......Page 368 12.44a The Helium Molecule-Ion He_2^+......Page 371 12.44b The Interaction of Two Normal Helium Atoms......Page 374 12.45 The One-Electron Bond, the Electron-Pair Bond, and the Three-Electron Bond......Page 375 13.46 Slater's Treatment of Complex Molecules......Page 379 13.46a Approximate Wave Functions for the System of Three Hydrogen Atoms......Page 381 13.46b Factoring the Secular Equation......Page 382 13.46c Reduction of Integrals......Page 383 13.46d Limiting Cases for the System of Three Hydrogen Atoms......Page 385 13.46e Generalization of the Method of Valence-Bond Wave Functions......Page 387 13.46f Resonance among Two or More Valence-Bond Structures......Page 390 13.46g The Meaning of Chemical Valence Formulas......Page 393 13.46h The Method of Molecular Orbitals......Page 394 14.47 Van der Waals Forces......Page 396 14.47a Van der Waals Forces for Hydrogen Atoms......Page 397 14.47c The Estimation of Van der Waals Forces from Molecular Polarizabilities......Page 400 14.48 The Symmetry Properties of Molecular Wave Functions......Page 401 14.48a Even and Odd Electronic Wave Functions. Selection Rules......Page 403 14.48b The Nuclear Symmetry Character of the Electronic Wave Function......Page 404 14.48c Summary of Results Regarding Symmetrical Diatomic Molecules......Page 407 14.49 Statistical Quantum Mechanics. Systems in Thermodynamic Equilibrium......Page 408 14.49a The Fundamental Theorem of Statistical Quantum Mechanics......Page 409 14.49b A Simple Application......Page 410 14.49c The Boltzmann Distribution Law......Page 412 14.49d Fermi-Dirac and Bose-Einstein Statistics......Page 415 14.49e The Rotational and Vibrational Energy of Molecules......Page 418 14.49f The Dielectric Constant of a Diatomic Dipole Gas......Page 421 14.50 The Energy of Activation of Chemical Reactions......Page 425 15.51 Matrix Mechanics......Page 429 15.51a Matrices and Their Relation to Wave Functions. The Rules of Matrix Algebra......Page 430 15.51b Diagonal Matrices and Their Physical Interpretation......Page 434 15.52 The Properties of Angular Momentum......Page 438 15.53 The Uncertainty Principle......Page 441 15.54 Transformation Theory......Page 445 Appendix I: Values of Physical Constants......Page 452 Appendix II: Proof That The Orbit Of A Particle Moving In A Central Field Lies In A Plane......Page 453 Appendix III: Proof Of Orthogonality Of Wave Functions Corresponding To Different Energy Levels......Page 454 Appendix IV: Orthogonal Curvlinear Coordinate Systems......Page 456 Appendix V: The Evaluation Of The Mutual Electrostatic Energy Of Two Spherically Symmetrical Distributions Of Electricity With Exponential Density Functions......Page 459 Appendix VI: Normalization Of The Associated Legendre Functions......Page 461 Appendix VII: Normalization Of The Associated Laguerre Functions......Page 464 Appendix VIII: The Greek Alphabet......Page 466 Index......Page 468 Back Cover......Page 483