Self-adjoint Extensions in Quantum Mechanics
General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials
(Sprache: Englisch)
After reviewing quantization problems emphasizing non-triviality of consistent operator construction, the book shows how problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems.
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After reviewing quantization problems emphasizing non-triviality of consistent operator construction, the book shows how problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems.
Klappentext zu „Self-adjoint Extensions in Quantum Mechanics “
This exposition is devoted to a consistent treatment of quantization problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces. The authors begin by considering quantization problems in general, emphasizing the nontriviality of consistent operator construction by presenting paradoxes to the naive treatment. It then builds the necessary mathematical background following it by the theory of self-adjoint extensions. By considering several problems such as the one-dimensional Calogero problem, the Aharonov-Bohm problem, the problem of delta-like potentials and relativistic Coulomb problemIt then shows how quantization problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems. In the end, related problems in quantum field theory are briefly introduced. This well-organized text is most suitable for students and post graduates interested in deepening their understanding of mathematical problems in quantum mechanics. However, scientists in mathematical and theoretical physics and mathematicians will also find it useful. 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
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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 and theoretical physics.
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 and theoretical physics.
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Inhaltsverzeichnis zu „Self-adjoint Extensions in Quantum Mechanics “
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.
Bibliographische Angaben
- Autoren: D.M. Gitman , I.V. Tyutin , B.L. Voronov
- 2012, 511 Seiten, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 0817644008
- ISBN-13: 9780817644000
- Erscheinungsdatum: 27.04.2012
Sprache:
Englisch
Rezension zu „Self-adjoint Extensions in Quantum Mechanics “
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)
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