Quantum Mechanics

I. Foundations of Quantum Theory.- 1. Energy and momentum of light quanta.- 2. Experimental test of the laws of conservation of energy and momentum for light quanta.- 3. Atomism.- 4. Bohr’s theory.- 5. The elementary quantum theory of radiation.- 6. Black-body radiation.- 7. De Broglie waves. The group velocity.- 8. Diffraction of microparticles.- II. Foundations of Quantum Mechanics.- 9. Statistical interpretation of de Broglie waves.- 10. The position probability of a microparticle.- 11. The principle of superposition of states.- 12. Momentum probability distribution of a microparticle.- 13. Mean values of functions of co-ordinates and functions of momenta.- 14. Statistical ensembles in quantum mechanics.- 15. The uncertainty relation.- 16. Illustrations of the uncertainty relation.- 17. The significance of the measuring apparatus.- III. Representation of Mechanical Quantities by Operators.- 18. Linear self-adjoint operators.- 19. The general formula for the mean value of a quantity and the mean square deviation.- 20. Eigenvalues and eigenfunctions of operators and their physical significance. ‘Quantisation’.- 21. Fundamental properties of eigenfunctions.- 22. General method of calculating the probabilities of the results of measurement.- 23. Conditions for a simultaneous measurement of different mechanical quantities to be possible.- 24. Co-ordinate and momentum operators of a microparticle.- 25. The angular momentum operator of a microparticle.- 26. The energy operator and the Hamilton’s function operator.- 27. The Hamiltonian.- IV. Change of State with Time.- 28. Schrödinger’s equation.- 29. Conservation of number of particles.- 30. Stationary states.- V. Change of Mechanical Quantities with Time.- 31. Time derivatives of operators.- 32. Equations of motion in quantum mechanics. Ehrenfest’s theorems.- 33. Integrals of the motion.- VI. The Relation Between Quantum Mechanics, Classical Mechanics and Optics.- 34. The transition from the quantum equations to Newton’s equations.- 35. The transition from Schrödinger’s time-dependent equation to the classical Hamilton-Jacobi equation.- 36. Quantum mechanics and optics.- 37. The quasiclassical approximation (the Wentzel-Kramers-Brillouin method).- VII. Basic Theory of Representations.- 38. Different representations of the state of quantum systems.- 39. Different representations of operators of mechanical quantities. Matrices.- 40. Matrices and operations on them.- 41. Determination of the mean value and spectrum of a quantity represented by an operator in matrix form.- 42. Schrödinger’s equation and the time dependence of operators in matrix form.- 43. Unitary transformations.- 44. The unitary transformation from one instant to another.- 45. The density matrix.- VIII. Theory of the Motion of Microparticles in a Field of Potential Forces.- 46. Introductory remarks.- 47. A harmonic oscillator.- 48. An oscillator in the energy representation.- 49. Motion in the field of a central force.- 50. Motion in a Coulomb field.- 51. The spectrum and wave functions of the hydrogen atom.- 52. Motion of an electron in univalent atoms.- 53. Currents in atoms. The magneton.- 54. Quantum levels of the diatomic molecule.- 55. Motion of an electron in a periodic field.- IX. Motion of a charged Microparticle in an Electromagnetic Field.- 56. An arbitrary electromagnetic field.- 57. Motion of a free charged particle in a uniform magnetic field.- X. Intrinsic Angular Momentum and Magnetic Moment of the Electron. Spin.- 58. Experimental proofs of the existence of electron spin.- 59. The electron spin operator.- 60. Spin functions.- 61. Pauli’s equation.- 62. Splitting of spectral lines in a magnetic field.- 63. Motion of the spin in a variable magnetic field.- 64. Properties of the total angular momentum.- 65. Labelling of atomic terms having regard to the electron spin. Multiplet structure of spectra.- XI. Perturbation Theory.- 66. Statement of the problem.- 67. Perturbation in the absence of degeneracy.- 68. Perturbation in the presence of degeneracy.- 69. Splitting of levels in the case of twofold degeneracy.- 70. Comments on the removal of degeneracy.- XII. Simple Applications of Perturbation Theory.- 71. The anharmonic oscillator.- 72. Splitting of spectral lines in an electric field.- 73. Splitting of spectral lines of the hydrogen atom in an electric field.- 74. Splitting of spectral lines in a weak magnetic field.- 75. A diagrammatic interpretation of the splitting of levels in a weak magnetic field (the vector model).- 76. Perturbation theory for the continuous spectrum.- XIII. Collision Theory.- 77. Statement of the problem in collision theory of microparticles.- 78. Calculation of elastic scattering by the Born approximation.- 79. Elastic scattering of fast charged microparticles by atoms.- 80. The exact theory of scattering. The phase shift of the scattered waves and the cross-section.- 81. The general case of scattering.- 82. Scattering of a charged particle in a Coulomb field.- XIV. Theory of Quantum Transitions.- 83. Statement of the problem.- 84. Transition probabilities under a time-dependent perturbation.- 85. Transitions due to a time-independent perturbation.- XV. Emission, Absorption and Scattering of Light by Atomic Systems.- 86. Introductory remarks.- 87. Absorption and emission of light.- 88. Emission and absorption coefficients.- 89. The correspondence principle.- 90. Selection rules for dipole radiation.- 91. Intensities in the emission spectrum.- 92. Dispersion.- 93. Raman scattering.- 94. Allowance for change of phase of the electromagnetic field of the wave within the atom. Quadrupole radiation.- 95. The photoelectric effect.- XVI. The Passage of Microparticles Through Potential Barriers.- 96. Statement of the problem and simplest cases.- 97. The apparent paradox of the ‘tunnel effect’.- 98. Cold emission of electrons from a metal.- 99. A three-dimensional potential barrier. Quasistationary states.- 100. The theory of a decay.- 101. Ionisation of atoms in strong electric fields.- XVII. The Many-Body Problem.- 102. General remarks on the many-body problem.- 103. The law of conservation of the total momentum of a system of microparticles.- 104. Motion of the centre of mass of a system of microparticles.- 105. The law of conservation of the angular momentum of a system of microparticles.- 106. Eigenfunctions of the angular momentum operator of the system. Clebsch-Gordan coefficients.- 107. The relation of the conservation laws to the symmetry of space and time.- XVIII. Simple Applications of the Theory of Motion of Many Bodies.- 108. Allowance for the motion of the nucleus in an atom.- 109. A system of microparticles executing small oscillations.- 110. Motion of an atom in an external field.- 111. Determination of the energy of stationary states of atoms from their deflection in an external field.- 112. Inelastic collisions between electrons and atoms. Determination of the energy of the stationary states of atoms by the collision method.- 113. The law of conservation of energy and the special significance of time in quantum mechanics.- XIX. Systems of Identical Microparticles.- 114. The identity of microparticles.- 115. Symmetric and antisymmetric states.- 116. Bose particles and Fermi particles. The Pauli principle.- 117. Wave functions for a system of fermions and bosons.- XX. Second Quantisation and Quantum Statistics.- 118. Second quantisation.- 119. The theory of quantum transitions and the second-quantisation method.- 120. The collision hypothesis. A Fermi-Dirac gas and a Bose-Einstein gas.- XXI. Multi-Electron Atoms.- 121. The helium atom.- 122. Approximate quantitative theory of the helium atom.- 123. The exchange energy.- 124. Quantum mechanics of the atom and Mendeleev’s periodic system of the elements.- XXII. Formation of Molecules.- 125. The hydrogen molecule.- 126. The nature of chemical forces.- 127. Dispersion forces between molecules.- 128. Nuclear spin in diatomic molecules.- XXIII. Magnetic Phenomena.- 129. Paramagnetism and diamagnetism of atoms.- 130. Ferromagnetism.- XXIV. The Atomic Nucleus.- 131. Nuclear forces. Isotopic spin.- 132. Systematics of states of a system of nucleons.- 133. Theory of the deuteron.- 134. Scattering of nucleons.- 135. Polarisation in the scattering of particles which have spin.- 136. The application of quantum mechanics to the systematics of elementary particles.- XXV. Conclusion.- 137. The formalism of quantum mechanics.- 138. The limits of applicability of quantum mechanics.- 139. Some epistemological problems.- Appendices.- I. The Fourier transformation.- II. Eigenfunctions when there is degeneracy.- III. Orthogonality and normalisation of eigenfunctions of the continuous spectrum. The ?-function.- IV. The significance of commutability of operators.- VI. Hamilton’s equations.- VII. Schrödinger’s equation and the equations of motion in curvilinear co-ordinates.- VIII. Conditions on the wave function.- IX. The solution of the oscillator equation.- X. An electron in a uniform magnetic field.- XI. Jacobi co-ordinates.- References.