Mechanics

Preface


1. The Lagrangian Formulation of Mechanics


1.01. The Harmonic Oscillator; a New Look at an Old Problem


1.02. A System and its Configuration


1.03. Generalized Coordinates and Velocities


1.04. Kinetic Energy and the Generalized Momenta


1.05. Lagrange's Equations


1.06. Holonomic Constraints


1.07. Electromagnetic Applications


1.08. Hamilton's Principle


2. The Hamiltonian Formulation Of Mechanics


2.01. Hamilton's Equations


2.02. The Hamiltonian as a Constant of the Motion


2.03. Hamiltonian Analysis of the Kepler Problem


2.04. Phase Space


3. Hamilton-Jacobi Theory


3.01. Canonical Transformations


3.02. Hamilton's Principal Function and the Hamilton-Jacobi Equation


3.03. Elementary Properties of Hamilton's Principal Function


3.04. Field Properties of Hamilton's Principal Function in the Context of Forced Motion


3.05. Hamilton's Principal Function and the Concept of Action


4. Waves


4.01. Waves on a String under Tension


4.02. Waves on a String under Tension and Local Restoring Force


4.03. The Superposition of Waves


4.04. Extension to Three Dimensions; Plane Waves


4.05. Quasi-Plane Waves; the Short Wavelength Limit


5. Historical Background of the Quantum Theory


5.01. Isothermal Cavity Radiation


5.02. Enumeration of Electromagnetic Modes; the Rayleigh-Jeans Result


5.03. Planck's Quantum Hypothesis


5.04. The Photoelectric Effect


5.05. Bohr's Explanation of the Hydrogen Spectrum


5.06. The Compton Effect


5.07. The de Broglie Relations and the Davisson-Germer Experiment


6. Wave Mechanics


6.01. The Two Branches of Quantum Theory


6.02. Waves and Wave Packets


6.03. The Schrodinger Equation


6.04. Interpretation of *P* W; Normalization and Probability Current


6.05. Expectation Values


7. The Time-Independent Schrodinger Equation and some of Its Applications


7.01. Time-independent Potential Energy Functions and Stationary Quantum States


7.02. The Rectangular Step; Transmission and Reflection


7.03. The Rectangular Barrier and Tunneling


7.04. Stationary States of the Infinite Rectangular Well


7.05. Stationary States of the Finite Rectangular Well; Bound States and Continuum States


7.06. The Particle in a Box


7.07. The One-dimensional Harmonic Oscillator


8. Operators, Observables, and the Quantization of a Physical System


8.01. General Definition of Operators; Linear Operators


8.02. The Non-commutative Algebra of Operators


8.03. Eigenfunctions and Eigenvalues; the Operators for Momentum and Position


8.04. The Association of an Operator with an Observable and the Calculation of Expectation Values


8.05. The Hamiltonian Operator and the Generalized Derivation of the Schrodinger Equation


8.06. Hermitian Operators and Expansion in Eigenfunctions


8.07. The Role of Hermitian Operators and their Eigenfunctions in Quantum Mechanics


9. The Significance of Expectation Values


9.01. Time Derivatives of Expectation Values


9.02. Ehrenfest's Theorem


9.03. A More Precise View of the Correspondence Principle and of the Nature of Classical Mechanics


10. The Momentum Representation


10.01. Fourier Series


10.02. Fourier Transforms and their Application to Quantum Mechanics


10.03. Extension to Three Dimensions


10.04. Eigenfunctions of Position and of Momentum


10.05. The Unforced Particle in the Momentum Representation


10.06. The Stationary State in the Momentum Representation


11. The Concept of Measurement in Quantum Mechanics


11.01. Measurements: Classical and Quantum


11.02. The Uncertainty Principle


11.03. Realization of the Minimum Uncertainty Product


12. The Hydrogenic Atom


12.01. Separation of Center-of-mass Motion from Relative Motion


12.02. Use of Spherical Polar Coordinates in the Analysis of the Relative Motion


12.03. Spherical Harmonics


12.04. Orbital Angular Momentum Operators


12.05. Solutions of the Radial Equation; Energy Levels


12.06. The Hydrogenic Wave Functions


13. Matrix Mechanics


13.01. The Non-commutative Algebra of Matrices


13.02. Matrix Formulation of Quantum Mechanics


13.03. Eigenvalues and Eigenvectors; the Diagonalization of a Matrix


13.04. Solution of a Quantum Mechanical Problem by Matrix Methods

Appendix A. Electromagnetic Interaction Energies in Terms of Local Potentials

Appendix B. Canonicity of the Transformation Generated By Gb(Qj, Pj, T)

Appendix C. Most Probable Distribution of Energy among Cavity Modes

Appendix D. Poisson Brackets

References

Selected Supplementary References

Problems

Name Index

Subject Index

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