Overview

Quantization in quantum mechanics deals with the problem of correct defining various classical structures, for example, quantum-mechanical observables such as Hamiltonian, momentum, self-adjoint operators in some Hilbert space and so on. Though there exists a naive treatment, based on experience in finite-dimensional algebra or even infinite-dimensional algebra with bounded operators, it results in paradoxes and inaccuracies. This exposition is devoted to a consistent treatment of such problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces.It begins by considering quantization problems in general, emphasizing the non-triviality 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 problem it 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.

Full Product Details

Author:   D.M. Gitman ,  I.V. Tyutin ,  B.L. Voronov
Publisher:   Birkhauser Boston Inc
Imprint:   Birkhauser Boston Inc
Edition:   2012 ed.
Volume:   62
Dimensions:   Width: 15.50cm , Height: 2.80cm , Length: 23.50cm
Weight:   0.951kg
ISBN:  

9780817644000

ISBN 10:   0817644008
Pages:   511
Publication Date:   27 April 2012
Audience:   College/higher education ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of stock  
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Table of Contents

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.- Schroedinger Operators with Exactly Solvable Potentials.- Dirac Operator with Coulomb Field.- Schroedinger and Dirac Operators with Aharonov-Bohm and Magnetic-Solenoid Fields.

Reviews

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)

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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