Operator Methods in Quantum Mechanics

1st Edition - January 1, 1981
Author: Martin Schechter
Language: English
eBook ISBN:
9 7 8 - 0 - 4 4 4 - 6 0 1 0 5 - 6

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,… Read more

Purchase options

LIMITED OFFER

Save 50% on book bundles

Immediately download your ebook while waiting for your print delivery. No promo code is needed.

Institutional subscription on ScienceDirect

Request a sales quote

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.

PrefaceAcknowledgmentsA Message to the ReaderList of SymbolsChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 8. Time-Independent Theory 8.1. The Resolvent Method 8.2. The Theory 8.3. A Simple Criterion 8.4. The Application ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 11. Oscillating Potentials 11.1. A Surprise 11.2. The Hamiltonian 11.3. The Estimates 11.4. A Variation 11.5. Examples ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 15. The Invariance Principle 15.1. Introduction 15.2. A Simple Result 15.3. The Estimates 15.4. An Extension 15.5. Another Form ExercisesChapter 16. Trace Class Operators 16.1. The Abstract Theorem 16.2. Some Consequences 16.3. Hilbert-Schmidt Operators 16.4. Verification for the Hamiltonian ExercisesAppendix A. The Fourier Transform Exercises AAppendix B. Hilbert Space Exercises BAppendix C. Holder's Inequality and Banach SpaceBibliographyIndex