A Development of Quantum Mechanics

1 / Quantization of Translatory Motion.- 1.1. Background Remarks on Time and Space.- 1.2. The Statistical Nature of Position, Velocity, and Momentum.- 1.3. A State Function Governing Translation.- 1.4. Diffraction by Neighboring Parallel Slits.- 1.5. Diffraction by Molecules.- 1.6. Dependence of the Function for a Definite Energy State on Time.- 1.7. Independent Movements within a State.- 1.8. The Continuum of Translational States.- 1.9. Periodic, Rectangularly Symmetric, Free Motion.- 1.10. Enumerating and Filling Translational States.- 1.11. Translational Energy of an Ideal Gas.- 1.12. Standing-Wave Translational Functions.- 1.13. Confined Rectangularly Symmetric Motion.- 1.14. Attenuated Motion.- 1.15. Joining Regions of Differing Potentials 3.- Discussion Questions.- Problems.- References.- 2 / Quantization of Rotatory Motion.- 2.1. Separation of Different Modes of Motion from Each Other.- 2.2. One-Particle Model for a Linear Rotator.- 2.3. Variation of ? in a Spherically Symmetric Field.- 2.4. The Rotational State Described by the Traveling Wave.- 2.5. Quantization of the Simple Rotator.- 2.6. Standing-Wave Rotational Functions.- 2.7. Energy Levels for a Two-Dimensional Rotator.- 2.8. Rotational Spectrum of a Linear Molecule.- 2.9. Broadening of Spectral Lines.- 2.10. Angular Momenta and the Resultant Energies for a Nonlinear Rotator.- 2.11. Quantizations of Prolate and Oblate Rotators.- 2.12. The Asymmetric Rotator.- 2.13. Dependence of ? on ? in a Spherically Symmetric Field.- 2.14. Explicit State Functions for Angular Motion in a Central Field.- Discussion Questions.- Problems.- References.- 3 / Quantization of Vibratory Motion.- 3.1. Additional Concerted Movements.- 3.2. Vibrational Kinetic and Potential Energies for a Diatomic Molecule.- 3.3. Derivatives of the State Function in Regions where the Potential Energy is not Constant.- 3.4. The Schrödinger Equation for Simple Harmonic Motion.- 3.5. Suitable State Functions for the Harmonic Oscillator.- 3.6. Properties of the Harmonic Oscillator ?v’s.- 3.7. Transitions between Vibrational Energy Levels.- 3.8. Vibrational Spectrum of a Diatomic Molecule.- 3.9. Anharmonicities and their Net Effects.- 3.10. Spectra of Polyatomic Molecules.- 3.11. Raman Spectra.- 3.12. The State Sum for a Vibrational Mode.- 3.13. Operator Formulations of the Harmonic Oscillator Equation.- Discussion Questions.- Problems.- References.- 4 / Radial Motion in a Coulombic Field.- 4.1. A More Complicated Oscillation.- 4.2. Representing the Axial and Angular Motions of Two Particles Bound Together.- 4.3. Differential Equations Governing Variations in a State Function.- 4.4. An Orthogonal-Coordinate Representation of ?2.- 4.5. Separating the Radial Variable from the Angular Variables in the Schrödinger Equation.- 4.6. The Radial Equation for an Electron — Nucleus System.- 4.7. Laguerre Polynomials.- 4.8. Quantization of the Radial Motion.- 4.9. Electronic Spectrum of a Hydrogen-like Atom or Ion.- 4.10. Uninuclear Multi-electron Structures.- 4.11. Useful Operator Formulations of the Radial Equation.- Discussion Questions.- Problems.- References.- 5 / Quantum Mechanical Operators.- 5.1. Dependence of Observables on the State Function.- 5.2. Eigenvalue Equations.- 5.3. Formula for the Expectation Value.- 5.4. The Dirac Delta.- 5.5. Operators for Momenta and Energy.- 5.6. The Hamiltonian Operator.- 5.7. Shift Operators for the Harmonic Oscillator.- 5.8. Nodeless Solutions.- 5.9. Acceptable Wavy Solutions.- 5.10. The Inverse Operator as the Adjoint of an Operator.- 5.11. Shift Operators for a Hydrogen-like Atom.- 5.12. Spontaneous Decay of an Unstable State.- Discussion Questions.- Problems.- References.- 6 / Wave Packets, Potentials, and Forces.- 6.1. Mixing Wave Functions.- 6.2. Fourier Series and Fourier Integrals.- 6.3. Broadening of a Level Caused by Instability.- 6.4. A Gaussian Wave Packet and its Wavevector Representation.- 6.5. An Arbitrary Superposition of Simple Planar Waves.- 6.6. The De Broglie Angular Frequency.- 6.7. Variations of a Wave Function and Observable Properties with Time.- 6.8. Generalizing Newton’s Laws.- 6.9. Scalar and Vector Potentials.- 6.10. The Lorentz Force Law.- 6.11. Phase Changes along a Given Path through a Field.- 6.12. Quantization of Magnetic Flux.- 6.13. Effect of a Field on Two-Slit Diffraction.- 6.14. Magnetic and Electric Aharonov—Bohm Effects.- Discussion Questions.- Problems.- References.- 7 / Angular Motion in a Spherically Symmetric Field.- 7.1. Conditions on the Angular Eigenfunctions.- 7.2. The Homogeneous Polynomial Factor.- 7.3. Solutions Derived from the Reciprocal of the Radius Vector.- 7.4. Dependence on the Azimuthal Angle.- 7.5. Dependence on the Colatitude Angle.- 7.6. Normalizing the Angular Eigenfunctions.- 7.7. Mutual Orthogonality of the Angular Eigenfunctions.- 7.8. Spherical-Harmonic Analysis of a Pure Planar Wave.- 7.9. Schrödinger Equation for Radial Constituents of Free Motion.- Discussion Questions.- Problems.- References.- 8 / Operators for Angular Momentum and Spin.- 8.1. Rotational States, Spin States, and Suitable Composites.- 8.2. The Azimuthal-Angle Operator Governing Angular Momentum.- 8.3. Cartesian Angular-Momentum Operators.- 8.4. Expressions Governing Rotational Standing Waves.- 8.5. Angular Momentum Shift Operators.- 8.6. The Integral or Half-Integral of Quantum Number M.- 8.7. Magnetic Energy Associated with an Angular Momentum.- 8.8. Observing Spatial Quantization in a Beam.- 8.9. Eigenoperators and Eigenfunctions for Spin.- 8.10. Relating State Functions for Different Magnetic Quantum Numbers.- 8.11. Combining Angular Momenta.- Discussion Questions.- Problems.- References.- 9 / Propagation, Spreading, and Scattering.- 9.1. The Current Associated with Propagation.- 9.2. An Evolution Operator.- 9.3. Spreading of a Gaussian Wave Packet.- 9.4. Existence of Scattering Centers in Materials.- 9.5. Cross Sections Presented by Scattering Centers.- 9.6. Pure Outgoing, Incoming, and Standing Spherical Waves.- 9.7. Possible Effects of Centers on the Partial Waves.- 9.8. Contributions to Various Cross Sections.- 9.9. A Classical Model for Scattering Centers.- 9.10. Nuclear Radii.- 9.11. Resonance.- 9.12. The Resonant State with Reaction.- 9.13. Singularity in the Partial Wave Amplitude Ratio at a Physical State.- Discussion Questions.- Problems.- References.- 10 / Investigating Multiparticle Systems.- 10.1. Quantum Mechanical Particles and Quasi Particles.- 10.2. The Variation Theorem.- 10.3. Separation of Particle Variables.- 10.4. Indistinguishability of Identical Particles in a Quantal System.- 10.5. Fundamental Combinatory Rules.- 10.6. Symmetrizing and Antisymmetrizing Operations.- 10.7. Basis for the Pauli Exclusion Principle.- 10.8. Relating the Energy of a Plural Paired-particle System to One- and Two-Particle Effects.- Discussion Questions.- Problems.- References.- Answers to Problems.- Name Index.