Quantum Mechanics in Hilbert Space

Name: Quantum Mechanics in Hilbert Space | Second Edition
Brand: Dover Publications
Price: 45.49 EUR
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Second Edition

Eduard Prugovecki(Autor*in)

Dover Publications (Verlag)

Erschienen am 2. Juli 2013

1195 Seiten

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Inhalt

Cover
Title Page
Copyright Page
Dedication
Contents
Preface to the Second Edition
Preface to the First Edition
List of Symbols
Introduction
I. Basic Ideas of Hilbert Space Theory
1. Vector Spaces
1.1 Vector spaces over fields of scalars
1.2 Linear independence of vectors
1.3 Dimension of a vector space
1.4 Isomorphism of vector spaces
Exercises
2. Euclidean (Pre-Hilbert) Spaces
2.1 Inner products on vector spaces
2.2 The concept of norm
2.3 Orthogonal vectors and orthonormal bases
2.4 Isomorphism of Euclidean spaces
Exercises
3. Metric Spaces
3.1 Convergence in metric spaces
3.2 Complete metric spaces
3.3 Completion of a metric space
Exercises
4. Hilbert Space
4.1 Completion of a Euclidean space
4.2 Separable Hilbert spaces
4.3 12 spaces as examples of separable Hilbert spaces
4.4 Orthonormal bases in Hilbert space
4.5 Isomorphism of separable Hilbert spaces
Exercises
5. Wave Mechanics of a Single Particle Moving in One Dimension
5.1 The formalism and its partial physical interpretation
5.2 The wave mechanical initial-value problem
5.3 Bound states of the system
5.4 A particle moving in a square-well potential
Exercises
References for Further Study
II. Measure Theory and Hilbert Spaces of Functions
*1. Measurable Spaces
1.1 Boolean algebras and cr algebras of sets
1.2 Boolean algebras of intervals
1.3 Borel sets in Rn
1.4 Monotone classes of sets
Exercises
*2. Measures and Measure Spaces
2.1 The concept of measure
2.2 Basic properties of measures
2.3 Extensions of measures and outer measures
2.4 Cartesian products of measure spaces
Exercises
*3. Measurable and Integrable Functions
3.1 The concept of a measurable function
3.2 Properties of measurable functions
3.3 Positive-definite integrable functions
3.4 Real and complex integrable functions
3.5 Infinite sequences and sums of integrals
3.6 Integration on Cartesian products of measure spaces
Exercises
4. Spaces of Square-Integrable Functions
4.1 Square-integrable functions
4.2 Hilbert spaces of square-integrable functions
*4.3 The separability of L2 spaces
4.4 Change of variables of integration
Exercises
5. The Hilbert Space of Systems of n Different Particles in Wave Mechanics
5.1 The Schroedinger equation of n-particle systems
5.2 The center-of-mass frame of reference
5.3 The bound states of n-particle systems
5.4 Properties of the w-particle Schroedinger operator
5.5 The initial-value problem
Exercises
6. Direct Sums and Tensor Products of Hilbert Spaces
6.1 Direct sums of Euclidean spaces
6.2 Separability and completeness of direct sums of Hilbert spaces
6.3 Bilinear forms on vector spaces
*6.4 Algebraic tensor products of vector spaces
*6.5 Hilbert tensor products of Hilbert spaces
Exercises
7. The Two-Body Bound-State Problem with a Spherically Symmetric Potential
7.1 Two particles interacting via a spherically symmetric potential
7.2 The equation of motion in spherical coordinates
7.3 Spherical harmonics on the unit sphere
7.4 The completeness of trigonometric functions
7.5 The completeness of Legendre polynomials
7.6 Completeness of the spherical harmonics
7.7 The two-body problem with a Coulomb potential
Exercises
References for Further Study
III. Theory of Linear Operators in Hilbert Spaces
1. Linear and Antilinear Operators on Euclidean Spaces
1.1 Linear and antilinear transformations
1.2 Algebraic operations with linear transformations
1.3 Continuous and bounded transformations
1.4 Examples of bounded and unbounded operators
Exercises
2. Linear Operators in Hilbert Spaces
2.1 Linear functionals on normed spaces
2.2 The dual of a Hilbert space
2.3 Adjoints of linear operators in Hilbert spaces
2.4 Bounded linear operators in Hilbert spaces
2.5 Dirac notation for linear operators
2.6 Closed operators and the graph of an operator
2.7 Nonexistence of unbounded everywhere-defined self-adjoint operators
Exercises
3. Orthogonal Projection Operators
3.1 Projectors onto closed subspaces of a Hilbert space
3.2 Algebraic properties of projectors
3.3 Partial ordering of projectors
3.4 Projectors onto intersections and orthogonal sums of subspaces
*3.5 Appendix: Extensions and adjoints of closed linear operators
Exercises
4. Isometric and Unitary Transformations
4.1 Isometric transformations in between Hilbert spaces
4.2 Unitary operators and the change of orthonormal basis
4.3 The Fourier-Plancherel transform
*4.4 Cayley transforms of symmetric operators
4.5 Self-adjointness of position and momentum operators in wave mechanics
Exercises
5. Spectral Measures
5.1 The point spectrum of a self-adjoint operator
5.2 Spectral resolution of self-adjoint operators with pure point spectrum
5.3 Weak, strong, and uniform operator limits
5.4 Spectral measures and complex measures
5.5 Spectral functions
*5.6 Appendix: Signed measures
Exercises
*6. The Spectral Theorem for Unitary and Self-Adjoint Operators
6.1 Spectral decomposition of a unitary operator
6.2 Monotonic sequences of linear operators
6.3 Construction of spectral families for unitary operators
6.4 Uniqueness of the spectral family of a unitary operator
6.5 Spectral decomposition of a self-adjoint operator
6.6 The spectral theorem for bounded self-adjoint operators
Exercises
References for Further Study
IV. The Axiomatic Structure of Quantum Mechanics
1. Basic Concepts in the Quantum Theory of Measurement
1.1 Observables and states in quantum mechanics
1.2 The concept of compatible observables
1.3 Born's correspondence rule for determinative measurements
1.4 Born's correspondence rule for preparatory measurements
1.5 The stochastic nature of the quantum theory of measurement
Exercises
2. Functions of Compatible Observables
2.1 Fundamental and nonfundamental observables
2.2 Bounded functions of commuting self-adjoint operators
2.3 Algebras of compatible observables
*2.4 Unbounded functions of commuting self-adjoint operators
Exercises
3. The Schroedinger, Heisenberg, and Interaction Pictures
3.1 The general form of the Schroedinger equation
3.2 The evolution operator
3.3 The Schroedinger picture
3.4 The Heisenberg picture and physical equivalence of formalisms
3.5 The formalism of matrix mechanics
3.6 The interaction picture
Exercises
4. State Vectors and Observables of Compound Systems
4.1 Superselection rules and state vectors
4.2 The Hilbert space of compound systems
4.3 Tensor products of linear operators
4.4 The observables of a system of distinct particles
4.5 Symmetric and antisymmetric tensor products of Hilbert spaces
4.6 The connection between spin and statistics
4.7 Spin and statistics for the n-body problem
Exercises
*5. Complete Sets of Observables
5.1 The concept of a complete set of operators
5.2 Cyclic vectors and complete sets of operators
5.3 The construction of spectral representation spaces
5.4 Cyclicity and maximality
Exercises
6. Canonical Commutation Relations
6.1 The empirical significance of commutation relations
6.2 Representations of canonical commutation relations
6.3 One-parameter Abelian groups of unitary operators
6.4 Representations of Weyl relations
*6.5 Appendix: Proof of von Neumann's theorem
Exercises
7. The General Formalism of Wave Mechanics
7.1 A derivation of one-particle wave mechanics
7.2 Wave mechanics of w-particle systems
7.3 The Schroedinger operator
7.4 Closures of linear operators
7.5 The Schroedinger kinetic energy operator
7.6 The Schroedinger potential energy operator
7.7 The self-adjointness of the Schroedinger operator
7.8 The angular momentum operators
**7.9 Time-dependent Hamiltonians
Exercises
*8. Completely Continuous Operators and Statistical Operators
8.1 Completely continuous operators
8.2 The trace of a linear operator
8.3 Hilbert-Schfnidt operators
8.4 The trace norm and the trace class
8.5 Statistical ensembles and the process of measurement
8.6 The quantum mechanical state of an ensemble
8.7 The von Neumann equation in Liouville space
8.8 Density matrices on spectral representation spaces
**8.9 Appendix: Classical and quantum statistical mechanics in master Liouville space
Exercises
References for Further Study
V. Quantum Mechanical Scattering Theory
1. Basic Concepts in Scattering Theory of Two Particles
1.1 Scattering theory and the initial-value problem
1.2 Asymptotic states in classical mechanics
1.3 Asymptotic states and scattering states in the Schroedinger picture
1.4 Moller wave operators
1.5 The scattering operator
1.6 The differential scattering cross section
1.7 The transition operator
*1.8 The T-matrix formula for the differential cross section
Exercises
2. General Time-Dependent Two-Body Scattering Theory
2.1 The intertwining property of wave operators
2.2 The partial isometry of wave operators
2.3 Properties of the S' operator
2.4 Initial and final domains of wave operators
2.5 Dyson's perturbation expansion
2.6 Criteria for existence of strong asymptotic states
2.7 The physical asymptotic condition
Exercises
3. General Time-Independent Two-Body Scattering Theory
3.1 The relation of the time-independent to the time-dependent approach
3.2 Lippmann-Schwinger equations in Hilbert space
3.3 Spectral integral representations of wave operators
3.4 The transition amplitude
3.5 The resolvent of an operator
3.6 The resolvent method in scattering theory
3.7 Appendix: Integration of vector- and operator-valued functions
**3.8 Appendix: Scattering theory in Liouville space
Exercises
4. Eigenfunction Expansions in Two-Body Potential Scattering Theory
4.1 Free plane waves in three dimensions
4.2 Distorted plane waves
4.3 Free and distorted spherical waves
4.4 Eigenfunction expansions for complete sets of operators
4.5 Green's operators and Green functions
4.6 Lippmann-Schwinger equations for eigenfunction expansions
4.7 The on-shell T-matrix
4.8 The off-shell T-matrix and 3T operators
**4.9 Appendix: Scattering theory for long-range potentials
**4.10 Appendix: Eigenfunctions and transition density matrices in statistical mechanics
Exercises
5. Green Functions in Potential Scattering
5.1 The free Green function
5.2 Partial wave free Green functions
5.3 Fredholm integral equations with Hilbert-Schmidt kernels
5.4 The full Green function
5.5 Fredholm expansion of the full Green function
5.6 Symmetry properties of the full Green function
*5.7 Appendix: The spectrum of the Schroedinger operator
*5.8 Appendix: Relations between resolvents and spectral functions
Exercises
6. Distorted Plane Waves in Potential Scattering
6.1 Potentials of Rollnik class
6.2 Fredholm series expressions for distorted plane waves
6.3 Asymptotic completeness and the generalized Parseval's equality
6.4 The scattering amplitude
6.5 The Born series
6.6 Distorted plane waves as solutions of the Schroedinger equation
*6.7 Appendix: Analytic opera tor-valued functions
Exercises
7. Wave and Scattering Operators in Potential Scattering
7.1 The existence of strong asymptotic states
7.2 The completeness of the Moller wave operators
*7.3 Proof of asymptotic completeness
7.4 Phase shifts for scattering in central potentials
7.5 The general phase-shift formula for the scattering operator
7.6 Partial-wave analysis for spherically symmetric potentials
Exercises
8. Fundamental Concepts in Multichannel Scattering Theory
8.1 The concept of channel
8.2 Channel Hamiltonians and wave operators
8.3 The uniqueness of channel strong asymptotic states
8.4 Interchannel scattering operators
8.5 The existence of strong asymptotic states in n-particle potential scattering
**8.6 Two-Hilbert space formulation of multichannel scattering theory
8.7 Multichannel eigenfunction expansions and T-matrices
**8.8 Multichannel Born approximations and Faddeev equations
*8.9 Appendix: von Neumann's mean ergodic theorem
Exercises
References for Further Study
Hints and Solutions to Exercises
References
Index

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