Operator Methods in Quantum Mechanics

Operator Methods in Quantum Mechanics demonstrates the power of operator theory as a tool in the study of quantum mechanics. More specifically, it shows how to use algebraic, representation-independent methods to solve one- and three-dimensional problems, including certain relativistic problems. It explains the applications of commutation relations, shift operators, and the virial, hypervirial, and Hellman-Feyman theorems to the calculation of eigenvalues, matrix elements, and wave functions. Organized into 16 chapters, this book begins by presenting a few simple postulates describing quantum theory and looking at a single particle moving along a straight line. Then, it introduces mathematical techniques that answer questions about the particle. It also discusses the use of spectral theorem in answering various questions concerning observables, along with negative eigenvalues and methods of determining parts of the spectrum or estimating lower bounds. Moreover, it explains the time-independent or stationary-state scattering theory and states, long-range potentials, and completeness and strong completeness. Oscillating potentials, eigenfunction expansions, restricted particles, hard-core potentials, the invariance principle, and the use of trace class operators to treat scattering theory are also described in this book. This volume is a valuable resource for physicists, as well as students of intermediate quantum mechanics and postgraduate students who want to be acquainted with the algebraic method of solving quantum mechanical problems.

Preface

Acknowledgments

A Message to the Reader

List of Symbols

Chapter 1. One-Dimensional Motion

1.1. Position

1.2. Mathematical Expectation

1.3. Momentum

1.4. Energy

1.5. Observables

1.6. Operators

1.7. Functions of Observables

1.8. Self-Adjoint Operators

1.9. Hilbert Space

1.10. The Spectral Theorem

Exercises

Chapter 2. The Spectrum

2.1. The Resolvent

2.2. Finding the Spectrum

2.3. The Position Operator

2.4. The Momentum Operator

2.5. The Energy Operator

2.6. The Potential

2.7. A Class of Functions

2.8. The Spectrum of H

Exercises

Chapter 3. The Essential Spectrum

3.1. An Example

3.2. A Calculation

3.3. Finding the Eigenvalues

3.4. The Domain of H

3.5. Back to Hilbert Space

3.6. Compact Operators

3.7. Relative Compactness

3.8. Proof of Theorem 3.7.5

Exercises

Chapter 4. The Negative Eigenvalues

4.1. The Possibilities

4.2. Forms Extensions

4.3. The Remaining Proofs

4.4. Negative Eigenvalues

4.5. Existence of Bound States

4.6. Existence of Infinitely Many Bound States

4.7. Existence of Only a Finite Number of Bound States

4.8. Another Criterion

Exercises

Chapter 5. Estimating the Spectrum

5.1. Introduction

5.2. Some Crucial Lemmas

5.3. A Lower Bound for the Spectrum

5.4. Lower Bounds for the Essential Spectrum

5.5. An Inequality

5.6. Bilinear Forms

5.7. Intervals Containing the Essential Spectrum

5.8. Coincidence of the Essential Spectrum with an Interval

5.9. The Harmonic Oscillator

5.10. The Morse Potential

Exercises

Chapter 6. Scattering Theory

6.1. Time Dependence

6.2. Scattering States

6.3. Properties of the Wave Operators

6.4. The Domains of the Wave Operators

6.5. Local Singularities

Exercises

Chapter 7. Long-Range Potentials

7.1. The Coulomb Potential

7.2. Some Examples

7.3. The Estimates

7.4. The Derivatives of V(x)

7.5. The Relationship Between Xt and V(x)

7.6. An Identity

7.7. The Reduction

7.8. Mollifiers

Exercises

Chapter 8. Time-Independent Theory

8.1. The Resolvent Method

8.2. The Theory

8.3. A Simple Criterion

8.4. The Application

Exercises

Chapter 9. Completeness

9.1. Definition

9.2. The Abstract Theory

9.3. Some Identities

9.4. Another Form

9.5. The Unperturbed Resolvent Operator

9.6. The Perturbed Operator

9.7. Compact Operators

9.8. Analytic Dependence

9.9. Projections

9.10. An Analytic Function Theorem

9.11. The Combined Results

9.12. Absolute Continuity

9.13. The Intertwining Relations

9.14. The Application

Exercises

Chapter 10. Strong Completeness

10.1. The More Difficult Problem

10.2. The Abstract Theory

10.3. The Technique

10.4. Verification for the Hamiltonian

10.5. An Extension

10.6. The Principle of Limiting Absorption

Exercises

Chapter 11. Oscillating Potentials

11.1. A Surprise

11.2. The Hamiltonian

11.3. The Estimates

11.4. A Variation

11.5. Examples

Exercises

Chapter 12. Eigenfunction Expansions

12.1. The Usefulness

12.2. The Problem

12.3. Operators on LP

12.4. Weighted LP-Spaces

12.5. Extended Resolvents

12.6. The Formulas

12.7. Some Consequences

12.8. Summary

Exercises

Chapter 13· Restricted Particles

13.1. A Particle Between Walls

13.2. The Energy Levels

13.3. Compact Resolvents

13.4. One Opaque Wall

13.5. Scattering on a Half-Line

13.6. The Spectral Resolution for the Free Particle on a Half-Line

Exercises

Chapter 14. Hard-Core Potentials

14.1. Local Absorption

14.2. The Modified Hamiltonian

14.3. The Resolvent Operator for H1

14.4. The Wave Operators W± (H1 H0)

14.5. Propagation

14.6. Proof of Theorem 14.5.1

14.7. Completeness of the Wave Operators W± , (H1 H0)

14.8. The Wave Operators W± (H, H1)

14.9. A Regularity Theorem

14.10. A Family of Spaces

Exercises

Chapter 15. The Invariance Principle

15.1. Introduction

15.2. A Simple Result

15.3. The Estimates

15.4. An Extension

15.5. Another Form

Exercises

Chapter 16. Trace Class Operators

16.1. The Abstract Theorem

16.2. Some Consequences

16.3. Hilbert-Schmidt Operators

16.4. Verification for the Hamiltonian

Exercises

Appendix A. The Fourier Transform

Exercises A

Appendix B. Hilbert Space

Exercises B

Appendix C. Holder's Inequality and Banach Space

Bibliography

Index