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Self-adjoint Extensions in Quantum Mechanics: General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials: Progress in Mathematical Physics, cartea 62

Autor D.M. Gitman, I.V. Tyutin, B.L. Voronov
en Limba Engleză Hardback – 27 apr 2012
Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and their spectral analysis.  Though a “naïve”  treatment exists for dealing with such problems, it is based on finite-dimensional algebra or even infinite-dimensional algebra with bounded operators, resulting in paradoxes and inaccuracies.   A proper treatment of these problems requires invoking certain nontrivial notions and theorems from functional analysis concerning the theory of unbounded self-adjoint operators and the theory of self-adjoint extensions of symmetric operators.
Self-adjoint Extensions in Quantum Mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naïve treatment.  The necessary mathematical background is then built by developing the theory of self-adjoint extensions.  Through examination of  various quantum-mechanical systems, the authors show how quantization problems associated with the correct definition of observables and their spectral analysis can be treated consistently for comparatively simple quantum-mechanical systems.  Systems that are examined include free particles on an interval, particles in a number of potential fields including delta-like potentials, the one-dimensional Calogero problem, the Aharonov–Bohm problem, and the relativistic Coulomb problem.
This well-organized text is most suitable for graduate students and postgraduates interested in deepening their understanding of mathematical problems in quantum mechanics beyond the scope of those treated in standard textbooks.  The book may also serve as a useful resource for mathematicians and researchers in mathematical andtheoretical physics.
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Specificații

ISBN-13: 9780817644000
ISBN-10: 0817644008
Pagini: 511
Ilustrații: XIII, 511 p. 3 illus.
Dimensiuni: 155 x 235 x 29 mm
Greutate: 0.91 kg
Ediția:2012
Editura: Birkhäuser Boston
Colecția Birkhäuser
Seria Progress in Mathematical Physics

Locul publicării:Boston, MA, United States

Public țintă

Research

Cuprins

Introduction.- Linear Operators in Hilbert Spaces.- Basics of Theory of s.a. Extensions of Symmetric Operators.- Differential Operators.- Spectral Analysis of s.a. Operators.- Free One-Dimensional Particle on an Interval.- One-Dimensional Particle in Potential Fields.- Schrödinger Operators with Exactly Solvable Potentials.- Dirac Operator with Coulomb Field.- Schrödinger and Dirac Operators with Aharonov-Bohm and Magnetic-Solenoid Fields.

Recenzii

From the reviews:
“In an infinite-dimensional Hilbert space a symmetric, unbounded operator is not necessarily self-adjoint. … The monograph by Gitman, Tyutin and Voronov is devoted to this problem. Its aim is to provide students and researchers in mathematical and theoretical physics with mathematical background on the theory of self-adjoint operators.” (Rupert L. Frank, Mathematical Reviews, February, 2013)

Textul de pe ultima copertă

Quantization of physical systems requires a correct definition of quantum-mechanical observables, such as the Hamiltonian, momentum, etc., as self-adjoint operators in appropriate Hilbert spaces and their spectral analysis.  Though a “naïve”  treatment exists for dealing with such problems, it is based on finite-dimensional algebra or even infinite-dimensional algebra with bounded operators, resulting in paradoxes and inaccuracies.   A proper treatment of these problems requires invoking certain nontrivial notions and theorems from functional analysis concerning the theory of unbounded self-adjoint operators and the theory of self-adjoint extensions of symmetric operators.
Self-adjoint Extensions in Quantum Mechanics begins by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes of the naïve treatment.  The necessary mathematical background is then built by developing the theory of self-adjoint extensions.  Through examination of  various quantum-mechanical systems, the authors show how quantization problems associated with the correct definition of observables and their spectral analysis can be treated consistently for comparatively simple quantum-mechanical systems.  Systems that are examined include free particles on an interval, particles in a number of potential fields including delta-like potentials, the one-dimensional Calogero problem, the Aharonov–Bohm problem, and the relativistic Coulomb problem.
This well-organized text is most suitable for graduate students and postgraduates interested in deepening their understanding of mathematical problems in quantum mechanics beyond the scope of those treated in standard textbooks.  The book may also serve as a useful resource for mathematicians and researchers in mathematical andtheoretical physics.

Caracteristici

Provides a consistent treatment of certain quantization problems in quantum mechanics with several examples Covers necessary mathematical background Clear organization Ends with a interesting discussion related to similar quantum field theory problems