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## Classical and Quantum

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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

Other Titles in the Series in Natural Philosophy

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1st Edition - January 1, 1976

Author: T. T. Taylor

Editor: D. ter Haar

eBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 8 7 3 4 - 1

Mechanics: Classical and Quantum explains the principles of quantum mechanics via the medium of analytical mechanics. The book describes Schrodinger's formulation, the… Read more

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Mechanics: Classical and Quantum explains the principles of quantum mechanics via the medium of analytical mechanics. The book describes Schrodinger's formulation, the Hamilton-Jacobi equation, and the Lagrangian formulation. The author discusses the Harmonic Oscillator, the generalized coordinates, velocities, as well as the application of the Lagrangian formulation to systems that are partially or entirely electromagnetic in character under certain conditions. The book examines waves on a string under tension, the isothermal cavity radiation, and the Rayleigh-Jeans result pertaining to the enumeration of electromagnetic modes. Other topics include Planck's quantum hypothesis and Bohr's explanation of the hydrogen spectrum. The book describes the two branches of quantum theory, namely. matrix mechanics ,and wave mechanics; it also covers other topics such as waves, wave packets and the Schrodinger equation. The book cites some applications of the time-independent Schrodinger equation; the author then analyzes the separation of center-of mass-motion from relative motion relating to the hydrogen atom. Nuclear physicists, scientists, and academicians in the field of nuclear physics or quantum mechanics will find this book highly valuable.

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

Other Titles in the Series in Natural Philosophy

- No. of pages: 412
- Language: English
- Published: January 1, 1976
- Imprint: Pergamon
- eBook ISBN: 9781483187341