Operator Methods in Quantum Mechanics

Operator Methods in Quantum Mechanics

1st Edition - January 1, 1981

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  • Author: Martin Schechter
  • eBook ISBN: 9780444601056

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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.

Table of Contents

  • Preface


    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


    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


    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


    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


    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


    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


    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


    Chapter 8. Time-Independent Theory

    8.1. The Resolvent Method

    8.2. The Theory

    8.3. A Simple Criterion

    8.4. The Application


    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


    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


    Chapter 11. Oscillating Potentials

    11.1. A Surprise

    11.2. The Hamiltonian

    11.3. The Estimates

    11.4. A Variation

    11.5. Examples


    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


    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


    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


    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


    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


    Appendix A. The Fourier Transform

    Exercises A

    Appendix B. Hilbert Space

    Exercises B

    Appendix C. Holder's Inequality and Banach Space



Product details

  • No. of pages: 346
  • Language: English
  • Copyright: © North Holland 1981
  • Published: January 1, 1981
  • Imprint: North Holland
  • eBook ISBN: 9780444601056

About the Author

Martin Schechter

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