Quantum Theory

Quantum Theory......Page 1 Title-Page......Page 3 Copyright......Page 4 PREFACE......Page 5 CONTENTS......Page 9 PART I - PHYSICAL FORMULATION OF THE QUANTUM THEORY......Page 13 1. Blackbody Radiation in Equilibrium......Page 17 3. Electromagnetic Potentials......Page 19 4. Boundary Conditions......Page 21 5. Fourier Analysis......Page 22 6. Polarization of Waves......Page 24 7. Evaluation of the Electromagnetic Energy......Page 25 8. Meaning of Preceding Result for Electromagnetic Energy......Page 26 9. Number of Oscillators......Page 27 10. Equipartition of Energy......Page 28 11. The Quantization of the Radiation Oscillators......Page 30 13. Material vs. Radiation Oscillators......Page 31 14. Quantization of Material Oscillators......Page 32 15. Summary......Page 34 1. Photoelectric Effect......Page 35 3. The Indivisibility of Quantum Processes......Page 38 4. Probability and Incomplete Determinism in Quantum Laws......Page 39 5. Unlikelihood of Completely Deterministic Laws on a Deeper Level......Page 41 6. Correspondence Principle......Page 42 7. Particle Properties of Light......Page 43 8. Compton Effect. The Scattering of Electromagnetic Radiation......Page 45 9. Analysis of the Compton Effect......Page 46 The Quantization of Material Systems......Page 49 10. Evidence for Quantization of All Material Systems......Page 50 11. Determination of Energy Levels......Page 51 13. Quantization of Angular Momentum......Page 53 14. The Hydrogen Atom......Page 54 Correspondence Theory of Radiation......Page 60 16. Absorption of Radiation......Page 61 17. Emission of Radiation......Page 66 Summary.......Page 70 1. Introduction......Page 71 2. Motion of Pulses of Light......Page 72 3. The Width of a Wave Packet......Page 74 4. Group Velocity......Page 75 5. Spread of Wave Packets......Page 77 7. Generalization to Three Dimensions......Page 79 8. Motion of Electron Wave Packets......Page 80 9. Effects of Forces......Page 81 10. Effects of Quantization......Page 82 12. Prediction of Electron Diffraction by Bohr-Sommerfeld Theory......Page 83 13. Interpretation of Wave Function in Terms of Probability......Page 85 14. Comparison between Electron Waves and Electromagnetic Waves......Page 86 15. More Detailed Picture of Electron Waves......Page 87 16. Transitions between Orbits......Page 88 17. Fourier Analysis. Fourier Integrals......Page 89 18. Wave Propagation for Free Particle.......Page 90 19. Wave Equation for Free Particle......Page 91 2. Choice of Probability Function P(x)......Page 93 3. Proof of Conservation of Probability......Page 94 4. Probability Current......Page 95 5. Is the Above Formulation the Most General One?......Page 96 6. Relativistic Theories......Page 101 7. The Probability Function for Light Quanta......Page 103 8. Probability of a Given Momentum......Page 104 9. The Relation between P(x) and P(k)......Page 107 10. Normalization Coefficient for P(k)......Page 108 Summary on Probabilities......Page 109 2. Proof of Uncertainty Principle for Electrons......Page 111 3. On the Interpretation of the Uncertainty Principle......Page 112 5. Relation of Stability of Atoms to Uncertainty Principle......Page 113 7. Modification of Measurements by Quantum Effects......Page 115 8. Microscope......Page 116 9. Measurement of Momentum......Page 117 11. Uncertainty Principle Applied to Light Quanta......Page 119 12. Observation of Light Quanta with Electron Microscope......Page 120 13. Localization of Electromagnetic Energy and Momentum by Means of Slits and Shutters......Page 123 14. Application of Uncertainty Principle to Problem of Defining Orbits in Atoms......Page 124 15. More General Application of Uncertainty Principle......Page 125 17. Are there Hidden Variables Underlying the Quantum Theory?......Page 126 1. The Interference Pattern, and the Wave-particle Nature of Matter......Page 128 2. Impossibility of Simultaneous Observation of Wave and Particle Properties of Matter......Page 130 3. Effects of Process of Observation on the Wave Function......Page 132 4. Relationship of Destruction of Interference to Consistency of Wave-particle Duality......Page 136 5. Generalization of Previous Results......Page 139 6. Measurement of Momentum......Page 141 8. Importance of Phase Relations......Page 143 9. Quantwn Properties of Matter as Potentialities......Page 144 11. On the Reality of the Wave Properties of Matter......Page 145 12. Wave-mechanical Interpretation of a Track in a Cloud Chamber......Page 149 13. Qualitative Picture of the Quantum Properties of Matter......Page 150 CHAPTER 7 - Summary of Quantum Concepts Introduced......Page 153 1. The Need for New Concepts......Page 156 3. Simple and Pictorial Ideas about Continuity of Motion......Page 157 5. Similarity of Simple Ideas about Fixed Position and Quantum Concepts......Page 158 6. More Sophisticated Ideas, Including Concept of Continuous Trajectory......Page 159 8. Early Ideas on Cause and Effect......Page 160 9. Completely Deterministic vs. Causal Laws as Tendencies......Page 162 10. Classical Theory Prescriptive and not Causal......Page 163 11. New Properties of Quantum Concepts: Approximate and Statistical Causality......Page 164 12. Energy and Momentum in Classical and Quantum Theories......Page 165 13. Momentum and Energy a Description of Causal Aspects of Matter......Page 167 14. Relation between Space Time and Causal Aspects of Matter......Page 168 15. The Principle of Complementarity......Page 170 16. The Indivisible Unity of the World......Page 173 17. Distinction between Object and Environment on Classical Level......Page 174 19. The Role of Causal Laws......Page 175 20. Analysis and Synthesis......Page 176 23. An Attempt to Analyze a Quantum System into Parts......Page 177 25. An Example: the Hydrogen Atom......Page 178 Summary of New Concepts in Quantum Theory......Page 179 Analogies to Ouantum Processes......Page 180 27. The Uncertainty Principle and Certain Aspects of Our Thought Processes......Page 181 28. Possible Reason for Analogies between Thought and Quantum Processes......Page 182 1. Wave Formalism and Probability......Page 185 2. Hypothesis of Linear Superposition......Page 186 3. Concept of the State of a System in Quantum Theory......Page 187 4.. Statistical Significance of the Concept of Quantum State......Page 188 Avarege Value of a Function of Position......Page 189 Criterion For Acceptable Wave Functions......Page 190 6. Operator Notation to Obtain Momentum Averages from Integrals in Position Space......Page 191 7. Functions of the Momentum......Page 192 8. Operators in Momentum Space. The Momentum Representation......Page 193 12. General Functions Expressed as Operators......Page 194 13. Reality of Average Values and the Order of Factors......Page 195 14. Hermitean Operators......Page 196 15. Modified Rule for Average of f(x, p)......Page 197 16. Hermitean Conjugate Operators......Page 198 17. Generalized Definition of a Hermitean Operator......Page 200 18. Generalized Definition of Hermitean Conjugates......Page 201 20. Application to Commutators......Page 202 22. General Form of Schrodinger's Equation......Page 203 24. Determination of H from Correspondence Principle......Page 204 25. General Formula for Time Derivative of Average Value of a Variable......Page 205 26. Application to Evaluation of Average Motion of Wave Packet......Page 206 29. Significance of Wave Equation......Page 208 32. Conservation of Energy......Page 209 2. Extension to Quantum Theory......Page 211 3. Correlations between p and x......Page 212 5. Specification of a Classical Statistical System through Mean Values of x^n p^m......Page 213 7. Application to Spreading Wave Packet for a Free Particle......Page 214 8. Semi-classical Picture of Particle with Uncertain Position and Momentum......Page 215 9. A Generalization of the Uncertainty Principle......Page 217 10. The Unusual Properties of the Gaussian Wave Function......Page 219 11. The Many-particle Problem......Page 220 12. Eigenvalues and Eigenfunctions of Operators......Page 221 13. Examples of Eigenfunctions and Eigenvalues in Position Space......Page 222 14. Degenerate Operators......Page 223 15. The Dirac Delta Function......Page 224 17. Momentum Representation of Eigenfunctions of x......Page 226 18. Connection Between Eigenfunctions of x in Position Space and in Momentum Space......Page 227 19. Differentiation of the Delta Function......Page 228 21. The Expansion of an Arbitrary Function as a Series of Eigenfunctions......Page 229 24. The Orthogonality of Eigenfunctions of a Hermitean Operator......Page 231 26. Expansion of Dirac Delta Function in Terms of Eigenfunctions of an Arbitrary Hermitean Operator......Page 232 27. Representation of an Operator in Terms of Its Eigenfunctions......Page 233 28. Mean Value of f(A) in Terms of Expansion into Eigenfunction of A......Page 234 30. Interference of Probabilities......Page 235 33. Change of Probability with Time. Stationary States......Page 237 34. Relation of Time-dependent Probabilities to Uncertainty Principle......Page 238 36. Change of Probability with Time for a General Wave Function......Page 239 1. Introduction to Part III......Page 241 3. Square Potentials......Page 242 4. Solution of Problem of Square Potential......Page 244 5. Penetration of a Barrier......Page 250 6. Applications of Barrier Penetration......Page 252 7. The Square Well Potential......Page 254 8. Width of Peak in Transmission Resonances......Page 257 9. The Ramsauer Effect......Page 258 10. Bound States......Page 259 11. Limit of an Infinitely Deep Well......Page 263 12. Graphical Interpretation of Solutions......Page 264 13. Application of Expansion Theorem......Page 265 14. Application to Deuteron......Page 266 16. Use of Observed Energy Levels to Provide Information about the Potential......Page 268 17. Wave Packets Made up from Eigenfunctions in the Continuum......Page 269 19. Time Delay of Wave as It Crosses Potential Well......Page 272 20. Metastable (or Virtual) States of Trapping an Object within a Well......Page 273 21. Metastable Singlet State of Deuteron......Page 274 1. Introduction......Page 276 2. The WKB Approximation......Page 277 4. Physical Interpretation of Solutions in Terms of Classical Distribution of Particles......Page 280 6. Wave Packets. The Time-dependent Solution......Page 281 6. Time-dependent Three-dimensional WKB Approximation......Page 282 8. Connection Formulas......Page 283 9. Connection Formulas......Page 285 10. Probability of Penetration of a Barrier......Page 287 11. Applications of Barrier Penetration Probability......Page 289 12. Probability of Penetration into Nucleus from Outside......Page 292 13. Bound States of a Potential Well......Page 293 14:. Virtual or Metastable States in the WKB Approximation......Page 295 15. Discussion of Eq. (64) for Transmissivity......Page 298 16. Width of Resonance. Transmission Coefficient near Resonance......Page 299 17. Intensity of Wave Inside Well......Page 300 18. Formation of Wave Packets. Lifetimes of Virtual States......Page 302 19. Wave Packet Inside Well (near Resonance)......Page 303 21. Application to Radioactive Systems......Page 305 22. Application to Nuclear Reactions......Page 306 2. Wave Equation......Page 308 3. General Form of Solutions......Page 309 6. Schrodinger's Method of Factorization......Page 310 6. Solution for Wave Functions......Page 312 7. Hermite Polynomials......Page 313 8. Normalization Factor......Page 314 10. Recurrence Relations......Page 315 12. General Form of Solution......Page 316 14. Expansion Postulate......Page 317 15. Wave Packets......Page 318 16. Mean Values of Kinetic and Potential Energies......Page 320 1. Separation of Variables......Page 322 2. Angular Momentum......Page 323 3. Commutation Rules for Angular Momentum......Page 324 5. Angular Momentum in Polar Co-ordinates......Page 325 7. Eigenvalues of Lz......Page 326 10. Determination of Simultaneous Eigenvalues and Eigenfunctions of Lz and L^2......Page 327 12. Effect of Fluctuation in Direction of L......Page 330 13. Eigenfunctions of L^2 and Lz......Page 331 14. Legendre Polynomials......Page 333 15. Associated Legendre Functions......Page 335 16. Measurement of Angular Momentum. Stern-Gerlach Experiment......Page 338 17. Transformation to a Rotated System of Axes......Page 339 19. Physical Equivalence of All Co-ordinate Systems......Page 342 21. Application to Construction of Orbits......Page 344 1. The Radial Equation......Page 346 4. Centrifugal Potential......Page 347 5. Separation into Relative Co-ordinates......Page 348 7. General Form of Solution for s Waves......Page 350 8. General Form of Solution When l > 0......Page 354 10. Physical Interpretation of Wave Functions of Different n, l, m......Page 355 12. Exact Solution for Hydrogen Atom......Page 357 13. Degeneracy of Hydrogen Energy Levels......Page 360 14. The Laguerre Polynomials and the Associated Laguerre Polynomials......Page 361 15. Three-dimensional Harmonic Oscillator......Page 363 19. An Important Property of Degenerate Eigenfunctions......Page 365 20. Relation of Hermite Polynomials to Spherical Harmonics......Page 366 21. The Hamiltonian for a Charged Particle in a Given Electromagnetic Field......Page 367 22. Quantum-mechanical Hamiltonian......Page 368 25. Gauge Invariance......Page 369 27. The Zeeman Splitting of Levels of Different m......Page 371 1. Matrix Representation of an Operator......Page 373 2. Properties of Matrices......Page 374 4. An Example: Harmonic-oscillator Wave Functions......Page 377 7. Commutation of Diagonal Matrices......Page 378 8. Continuous Matrices......Page 379 9. Column Representation of the Wave Function......Page 380 10. Normalization and Orthogonality of Wave Functions in Column Representation......Page 381 12. Eigenvalues and Eigenvectors of Matrices......Page 382 13. Change of Representation......Page 383 15. Significance of the Unitary Transformation......Page 384 17. Simultaneous Eigenfunctions of Commuting Operators......Page 387 18. The Specification of an Arbitrary Operator in Terms of Its Commutators with a Complete Commuting Set of Operators......Page 389 19. Schrodinger's Equation in an Arbitrary Representation......Page 390 21. The Heisenberg Representation......Page 391 22. Time Rate of Change of Operators in a Heisenberg Representation......Page 392 23. Poisson Brackets......Page 394 24. Heisenberg's Formulation of Quantum Theory......Page 395 25. Physical Interpretation of Matrix Representations and Transformation Theory......Page 396 1. Electron Spin......Page 399 2. Matrix Representation of Angular-momentum Operators......Page 400 3. The Allowed Values of l and m; Half-integral Angular-momentum Quantum Numbers......Page 401 4. Matrices for (Lx + i Ly) and (Lx - i Ly)......Page 402 5. The Eigenfunctions of the sigma Operators......Page 404 7. Spinor Transformations......Page 405 8. The Addition of Angular Momenta......Page 409 9. Addition of Spin Angular Momenta of Two Separate Particles......Page 410 10. Probability Distribution of Spin States in a Statistical Ensemble......Page 412 11. Addition of Orbital and Spin Angular Momenta of a Given Particle......Page 413 12. Discussion of General Problem of Adding Angular Momenta......Page 416 13. Energy of a Spinning Electron......Page 417 2. Case of a Small Perturbation (Method of Variation of Constants)......Page 419 4. Methods of Approximation......Page 422 5. Interpretation of the |Cm|^2 in Terms of Transition Probabilities......Page 423 7. Case a: Vmn Turned on Suddenly (Calculation to First Order in lambda)......Page 424 9. A Description of Transitions in Terms of Quantum Fluctuations......Page 425 10. Microscopic Reversibility of Transition Processes......Page 427 11. Conservation of Probability......Page 428 12. Case b: Trigonometric Variation of Vmn with Time, with Application to Absorption and Emission of Light......Page 429 14. Interpretation of Results......Page 430 15. Present Treatment Does Not Quantize Radiation Field......Page 431 17. Relation between Vector Potential and Intensity......Page 432 18. Effect of Distribution of Frequencies of the Incident Light Wave......Page 433 20. Induced Emission of Quanta......Page 435 23. Einstein's Treatment of Spontaneous Emission......Page 436 24. Applications of Transition Theory......Page 438 25. Electric Dipole Approximation......Page 439 26. Evaluation of alpha_nm for Isotropic Harmonic Oscillator......Page 441 27. Selection Rules for Harmonic Oscillator......Page 442 29. Introduction of Parity......Page 443 30. Selection Rules on Parity......Page 444 31. Selection Rules for Spherically Symmetric Potential, with the Neglect of Spin......Page 445 32. Forbidden Transitions, Electric Quadripole Radiation......Page 447 34. Selection Rules for Electric Quadripole Radiation......Page 448 36. Higher Order Transitions......Page 449 38. Total Rate of Radiation......Page 450 39. Comparison with Classical Theory......Page 451 41. Circular Polarization......Page 453 43. Quantum Treatment......Page 454 45. Application to Normal Zeeman Effect......Page 455 46. Quantum Description of Normal Zeeman Effect......Page 457 49. Effects of Electron Spin on Transition Probabilities......Page 458 50. Case c: Vmn Varies Slowly with the Time (Adiabatic Case).......Page 460 51. Adiabatic Turning on of Potential Results in Perturbed Stationary......Page 462 52. Perturbation of Stationary-state Wave Functions......Page 465 53. Interpretation of Second-order Fonnulas for Energy......Page 468 54. Application of Perturbation Theory......Page 470 1. Introduction......Page 474 2. Example: Doubly Degenerate Level......Page 475 5. Higher Approximations......Page 476 6. More Than Two Degenerate Levels......Page 477 8. Time-dependent Solution for Special Case of Two Degenerate Levels......Page 478 9. Quantum-mechanical "Resonance"......Page 479 10. Analogy of Degeneracy Problem to Principal Axis Transformation......Page 481 11. First-order Stark Effect......Page 482 12. Classical Interpretation of Linear Stark Etlect......Page 483 13. Van der Waals Forces between Atoms......Page 484 14. Quantum-mechanical Analogue of Oscillator Phase......Page 490 16. Experimental Consequences of Degeneracy......Page 491 16. Exchange Degeneracy......Page 492 17. Solution of Problem......Page 493 19. Evaluation of the Energy......Page 494 20. Higher Approximations......Page 497 21. Effects of Spin......Page 498 23. Correlation between Exchange Energy and Electron Spin Brought about by Antisymmetry of Wave Functions......Page 500 25. A System of Many Electrons......Page 502 26. Pauli Exclusion Principle......Page 503 28. Totally Symmetric Wave Functions......Page 504 29. Indistinguishability of Equivalent Particles......Page 505 1. General Adiabatic Perturbations......Page 508 2. Interpretation of Results......Page 512 3. Applications......Page 513 4. The Approximation of Sudden Change of Potential......Page 519 5. Application. Emission of Electron from Nucleus in Beta-Decay......Page 520 6. Relation between Perturbation Theory and Theory of Sudden Transitions......Page 521 2. Classical Theory of Scattering......Page 523 3. Definition of Cross Section......Page 524 5. Cross Section as a Function of Scattering Angle......Page 525 6. Difierential Cross Sections......Page 526 7. More General Theory of Scattering......Page 528 8. The Approximation of Small Deflections. Classical Perturbation Theory......Page 529 9. Cross Section for Energy and Momentum Transfer......Page 532 10. Exact Solution for Scattering......Page 533 11. Use of Cross Sections to Investigate Law of Force......Page 535 12. Transformation from Center-of-Mass System to Laboratory System of Co-ordinates......Page 536 15. Quantum Theory of Scattering......Page 539 16. Condition for Validity of Classical Theory of Scattering......Page 540 17. Quantum Description of Scattering......Page 541 18. Scattering Considered as a Transition between Different States in Momentum Space......Page 543 19. Born Approximation. Perturbation Theory......Page 545 20. Evaluation of Cross Section......Page 548 21. Example of Application: The Shielded Coulomb Force......Page 549 22. Relation between Born Approximation and Fourier Analysis of the Potential......Page 550 23. Illustration: Comparison of Cross Sections for Gaussian Potential and Square Well......Page 551 24. The Space-time Representation of Scattering......Page 553 25. New Form for Schrodinger's Equation......Page 555 26. Interpretation of Results......Page 558 28. Relation of Space-time and Causal Descriptions......Page 559 29. Relation of Stationary-state Method to Time Dependent Descriptions......Page 560 30. Another Application of the Bom Approximation: Scattering from a Crystal Lattice......Page 562 31. Conditions for Validity of Born Approximation......Page 563 32. Application to Screened Coulomb Scattering......Page 564 33. Another Criterion for Validity of Born Approximation......Page 565 36. Unusual Properties of Coulomb Force......Page 566 37. Lack of Applicability of Born Approximation to Nuclei......Page 567 38. Application to Shielded Coulomb Force......Page 568 39. Method of Partial Waves. (Rayleigh, Faxen and Holtsmark)......Page 569 41. Special Case: Coulomb Potential......Page 570 44. Interpretation of Partial Waves......Page 572 45. Boundary Conditions on Partial Waves for Free Particle......Page 573 46. Imposition of Boundary Conditions When a Potential is Present......Page 575 47. Formula for Scattering Cross Section......Page 576 48. Total Cross Section......Page 577 49. Calculation of Phase for Impenetrable Sphere......Page 578 50. Application of Enct Method to Scattering from Square Well for s Waves......Page 579 51. Ramsauer Effect......Page 580 53. Application to Nuclear Scattering......Page 582 54. Approximate Expression of Low-energy Cross Section in Terms of Binding Energy of Deuteron......Page 583 55. Spin-dependent Forces......Page 584 56. Solution for Depth of Single Well......Page 586 57. Comparison with Experiment; Measurements of Radius of Potential......Page 587 58. Coulomb Scattering......Page 589 59. Interpretation of Above Result......Page 590 60. Exchange Effects in Coulomb Scattering......Page 591 2. The Nature of the Observing Apparatus......Page 595 3. The Classical Stages of an Observing Apparatus......Page 596 4. Extent of Arbitrariness in Distinction Between the Observer and What He Sees......Page 598 5. Mathematical Treatment of Process of Observation......Page 600 6. An Example: The Measurement of the Spin of an Atom......Page 605 7. Generalization to a Variable with an Arbitrarily Large Number of Eigenvalues......Page 610 8. Destruction of Interference in the Process of Measurement......Page 612 10. Interpretation of Combined Wave Function in Terms of a Statistical Ensemble of Wave Functions for the Spin Alone......Page 614 11. Inclusion of Apparatus Co-ordinates......Page 616 12. Irreversibility of Process of Measurement and Its Fundamental Role in Quantum Theory......Page 620 13. Wave vs. Particle Properties of Matter as Potentialities......Page 621 14. On the Relation between Continuity and Discontinuity in Quantum Transfers......Page 622 15. The Paradox of Einstein, Rosen, and Podolsky......Page 623 16. The Hypothetical Experiment of Einstein, Rosen, and Podolsky......Page 626 17. Mathematical Analysis of Experiment According to Quantum Theory......Page 627 18. Physical Description of Origin of Correlations......Page 631 19. Proof that Quantum Theory Is Inconsistent with Hidden Variables......Page 634 CHAPTER 23 - Relationship between Quantum and Classical Concepts......Page 636 INDEX......Page 641 Back Cover......Page 660