# Mechanics

### Classical and Quantum

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Mechanics: Classical and Quantum is a 13-chapter book that begins by explaining the Lagrangian and Hamiltonian formulation of mechanics. The Hamilton-Jacobi theory, historical background of the quantum theory, and wave mechanics are then described. Subsequent chapters discuss the time-independent Schrödinger equation and some of its applications; the operators, observables, and the quantization of a physical system; the significance of expectation values; and the concept of measurement in quantum mechanics. The matrix mechanics and the "hydrogenic atom", an atom in which one electron moves under the influence of a nucleus of charge that, to a very good approximation, can be thought of as a point, are also presented. This book will be very useful to students studying this field of interest.

## Table of Contents

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 Schrödinger Equation

6.04. Interpretation of Ψ*Ψ; Normalization and Probability Current

6.05. Expectation Values

7. The Time-Independent Schrödinger 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 Schrödinger 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

Other Titles in the Series in Natural Philosophy

## Product details

- No. of pages: 412
- Language: English
- Copyright: © Pergamon 2013
- Published: January 1, 1976
- Imprint: Pergamon
- eBook ISBN: 9781483139982