Spectral Theory and Quantum Mechanics

Spectral Theory and Quantum Mechanics

This book discusses the mathematical foundations of quantum theories.

Author: Valter Moretti

Publisher: Springer

ISBN: 9783319707068

Category: Mathematics

Page: 950

View: 786

This book discusses the mathematical foundations of quantum theories. It offers an introductory text on linear functional analysis with a focus on Hilbert spaces, highlighting the spectral theory features that are relevant in physics. After exploring physical phenomenology, it then turns its attention to the formal and logical aspects of the theory. Further, this Second Edition collects in one volume a number of useful rigorous results on the mathematical structure of quantum mechanics focusing in particular on von Neumann algebras, Superselection rules, the various notions of Quantum Symmetry and Symmetry Groups, and including a number of fundamental results on the algebraic formulation of quantum theories. Intended for Master's and PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book also benefits established researchers by organizing and presenting the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly."
Categories: Mathematics

Intermediate Spectral Theory and Quantum Dynamics

Intermediate Spectral Theory and Quantum Dynamics

The spectral theory of linear operators in Hilbert spaces is the most important tool
in the mathematical formulation of quantum mechanics; in fact, linear operators
and quantum mechanics have had a symbiotic relationship. However, typical ...

Author: César R. de Oliveira

Publisher: Springer Science & Business Media

ISBN: 3764387955

Category: Science

Page: 410

View: 608

The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics. Many examples and exercises are included that focus on quantum mechanics.
Categories: Science

Spectral Theory and Partial Differential Equations

Spectral Theory and Partial Differential Equations

Conference in Honor of James Ralston's 70th Birthday on Spectral Theory and
Partial Differential Equations: June 17--21, 2013, ... MR2786226 (2012h:81287)
V. Georgescu and C. Gérard, On the virial theorem in quantum mechanics,
Comm.

Author: James V Ralston

Publisher: American Mathematical Soc.

ISBN: 9781470409890

Category: Differential equations, Partial

Page: 197

View: 202

This volume contains the proceedings of the Conference on Spectral Theory and Partial Differential Equations, held from June 17-21, 2013, at the University of California, Los Angeles, California, in honor of James Ralston's 70th Birthday. Papers in this volume cover important topics in spectral theory and partial differential equations such as inverse problems, both analytical and algebraic; minimal partitions and Pleijel's Theorem; spectral theory for a model in Quantum Field Theory; and beams on Zoll manifolds.
Categories: Differential equations, Partial

A Spectral Theory of Time dependent Quantum Mechanics

A Spectral Theory of Time dependent Quantum Mechanics

It is quite fitting that modern functional analysis , which has a strong root in the
spectral theory of time - independent quantum mechanics , now provides a
natural setting for the time - dependent problem . The main purpose of this work
is to ...

Author: Joyce Mary Okuniewicz

Publisher:

ISBN: MINN:31951D018520451

Category:

Page: 268

View: 508

Categories:

Spectral Theory of Schr dinger Operators

Spectral Theory of Schr  dinger Operators

Such an inference is an example of the great potential which lies in numerical
spectral analysis to obtain insight into the ... References [1] Amrein, W. O., Jauch,
J. M., Sinha, K. B., Scattering Theory in Quantum Mechanics, Benjamin, Reading
 ...

Author: Rafael del Río

Publisher: American Mathematical Soc.

ISBN: 9780821832974

Category: Mathematics

Page: 249

View: 229

This volume gathers the articles based on a series of lectures from a workshop held at the Institute of Applied Mathematics of the National University of Mexico. The aim of the book is to present to a non-specialized audience the basic tools needed to understand and appreciate new trends of research on Schrodinger operator theory. Topics discussed include various aspects of the spectral theory of differential operators, the theory of self-adjoint operators, finite rank perturbations, spectral properties of random Schrodinger operators, and scattering theory for Schrodinger operators. The material is suitable for graduate students and research mathematicians interested in differential operators, in particular, spectral theory of Schrodinger operators.
Categories: Mathematics

Hilbert Space Methods in Quantum Mechanics

Hilbert Space Methods in Quantum Mechanics

The necessary foundation in quantum mechanics is covered in this book.

Author: Werner O. Amrein

Publisher: EPFL Press

ISBN: 1420066811

Category: Science

Page: 395

View: 803

The necessary foundation in quantum mechanics is covered in this book. Topics include basic properties of Hibert spaces, scattering theory, and a number of applications such as the S-matrix, time delay, and the Flux-Across-Surfaces Theorem.
Categories: Science

Quaternionic Quantum Mechanics and Quantum Fields

Quaternionic Quantum Mechanics and Quantum Fields

In much of what follows there is a close analogy with the familiar framework of
complex quantum mechanics, but there are a number of characteristic ... Second,
although the spectral theory for quaternion self-adjoint operators (see Sec.

Author: Stephen L. Adler

Publisher: Oxford University Press

ISBN: 0195345061

Category: Science

Page: 608

View: 429

It has been known since the 1930s that quantum mechanics can be formulated in quaternionic as well as complex Hilbert space. But systematic work on the quaternionic extension of standard quantum mechanics has scarcely begun. Authored by a world-renowned theoretical physicist, this book signals a major conceptual advance and gives a detailed development and exposition of quaternionic quantum mechanics for the purpose of determining whether quaternionic Hilbert space is the appropriate arena for the long sought-after unification of the standard model forces with gravitation. Significant results from earlier literature, together with many new results obtained by the author, are integrated to give a coherent picture of the subject. The book also provides an introduction to the problem of formulating quantum field theories in quaternionic Hilbert space. The book concludes with a chapter devoted to discussions on where quaternionic quantum mechanics may fit into the physics of unification, experimental and measurement theory issues, and the many open questions that still challenge the field. This well-written treatise is a very significant contribution to theoretical physics. It will be eagerly read by a wide range of physicists.
Categories: Science

Spectral Theory and Its Applications

Spectral Theory and Its Applications

The development of spectral theory is strongly related to quantum mechanics,
and the main operators that immediately appear in the theory are the operators of
multiplication by x (in, say, L2 (11%)), the operator of differentiation d /dx, and the
 ...

Author: Bernard Helffer

Publisher: Cambridge University Press

ISBN: 9781107032309

Category: Mathematics

Page: 255

View: 528

Introduces the basic tools in spectral analysis using numerous examples from the Schrödinger operator theory and various branches of physics.
Categories: Mathematics

Differential Operators and Spectral Theory

Differential Operators and Spectral Theory

Mathematical results in quantum mechanics (Blossin, 1993). Operator Theory
Adv. Appl., vol. 70, Birkhauser, Basel, 1994, pp. 3-7. [1 19] Birman, M. Sh., Weidl,
T. The discrete spectrum in a gap of the continuous one for compact supported ...

Author: M. Sh Birman

Publisher: American Mathematical Soc.

ISBN: 0821813870

Category: Mathematics

Page: 285

View: 822

This volume contains a collection of original papers in mathematical physics, spectral theory, and differential equations. The papers are dedicated to the outstanding mathematician, Professor M. Sh. Birman, on the occasion of his 70th birthday. Contributing authors are leading specialists and close professional colleagues of Birman. The main topics discussed are spectral and scattering theory of differential operators, trace formulas, and boundary value problems for PDEs. Several papers are devoted to the magnetic Schrodinger operator, which is within Birman's current scope of interests and recently has been studied extensively. Included is a detailed survey of his mathematical work and an updated list of his publications. This book is aimed at graduate students and specialists in the above-mentioned branches of mathematics and theoretical physicists. The biographical section will be of interest to readers concerned with the scientific activities of Birman and the history of those branches of analysis and spectral theory where his contributions were important and often decisive. Features: The first detailed survey of Birman's mathematical work; includes an updated bibliography. New material on the history of some branches of analysis. Prominent authors: Lieb, Agmon, Deift, Simon, Ladyzhenskaya, and others. All original works, containing new results in fields of great current interest.
Categories: Mathematics

Fourth Summer School in Analysis and Mathematical Physics

Fourth Summer School in Analysis and Mathematical Physics

Topics in Spectral Theory and Quantum Mechanics, May 2005, Universidad
Nacional Autónoma de México, Cuernavaca, Mexico ... Quantum mechanics
started at the beginning of the last century as a revolutionary theory to explain
certain ...

Author: Carlos Villegas-Blas

Publisher: American Mathematical Soc.

ISBN: 9780821840641

Category: Mathematics

Page: 148

View: 774

This book consists of three expository articles written by outstanding researchers in Mathematical Physics: Rafael Benguria, Peter Hislop, and Elliott Lieb. The articles are based on their lectures at the Fourth Summer School in Analysis and Mathematical Physics, held at the Institute of Mathematics, Universidad Nacional Autonoma de Mexico, Cuernavaca in May 2005. The main goal of the articles is to link the basic knowledge of a graduate student in Mathematics with three current research topics in Mathematical Physics: Isoperimetric inequalities for eigenvalues of the Laplace Operator, Random Schrodinger Operators, and Stability of Matter, respectively. These well written articles will guide and introduce the reader to current research topics and will also provide information on recent progress in some areas of Mathematical Physics.
Categories: Mathematics

Scattering Theory in Quantum Mechanics

Scattering Theory in Quantum Mechanics

It is devoted to the study of certain structural properties of self - adjoint operators
that are very useful in quantum mechanics ( cf . Section 3 - 2 ) . The basic
theorem is the socalled spectral theorem which in an appropriate sense
generalizes the ...

Author: Werner O. Amrein

Publisher: Addison Wesley Longman

ISBN: UCAL:B4513680

Category: Quantum theory

Page: 691

View: 183

Categories: Quantum theory

The Logico Algebraic Approach to Quantum Mechanics

The Logico Algebraic Approach to Quantum Mechanics

SPECTRAL THEORY IN QUANTUM LOGICS ABSTRACT. If one supposes a
quantum logic L to be a <j-orthocomplete, orthomodular partially ordered set
admitting a set of a-orthoadditive functions (called states) from L to the unit
intervals [0, ...

Author: C.A. Hooker

Publisher: Springer Science & Business Media

ISBN: 9027707073

Category: Philosophy

Page: 466

View: 717

The twentieth century has witnessed a striking transformation in the understanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi tional opinion that one should look to the specific applications of a theory in orrter to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that structure, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical manoeuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrodinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation, the elementary theory moved, flanked even at this later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic alternative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical structures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manipulation of purely abstract structures.
Categories: Philosophy

Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators

Geometric and Arithmetic Methods in the Spectral Theory of Multidimensional Periodic Operators

The fundamental result of the spectral theory of periodic operators is the
assertion that the spectrum of the original operator ... Serious interest in
multidimensional periodic operators arose under the influence of quantum
mechanics after it was ...

Author: M. M. Skriganov

Publisher: American Mathematical Soc.

ISBN: 0821831046

Category: Mathematics

Page: 121

View: 145

Categories: Mathematics

Quantum Theory

Quantum Theory

In the early development of quantum mechanics in two different forms, as 'matrix
mechanics' by Heisenberg and as 'wave ... 21.10 the centre piece of von
Neumann's Hilbert space formalism for quantum mechanics, the spectral theorem
for ...

Author: Peter Bongaarts

Publisher: Springer

ISBN: 9783319095615

Category: Science

Page: 445

View: 734

This book was inspired by the general observation that the great theories of modern physics are based on simple and transparent underlying mathematical structures – a fact not usually emphasized in standard physics textbooks – which makes it easy for mathematicians to understand their basic features. It is a textbook on quantum theory intended for advanced undergraduate or graduate students: mathematics students interested in modern physics, and physics students who are interested in the mathematical background of physics and are dissatisfied with the level of rigor in standard physics courses. More generally, it offers a valuable resource for all mathematicians interested in modern physics, and all physicists looking for a higher degree of mathematical precision with regard to the basic concepts in their field.
Categories: Science

Quantum Mechanics and Quantum Field Theory

Quantum Mechanics and Quantum Field Theory

By the spectral theorem (see problem 1.15) (ψ,Aψ) = ∫ λd(ψ,E(λ)ψ) (3.4) As in
classical probability theory, (ψ,Aψ) is interpreted as the average value of
repeated measurements. It is called the expectation value of the observable A. If
we have ...

Author: Jonathan Dimock

Publisher: Cambridge University Press

ISBN: 9781139497480

Category: Science

Page:

View: 909

Explaining the concepts of quantum mechanics and quantum field theory in a precise mathematical language, this textbook is an ideal introduction for graduate students in mathematics, helping to prepare them for further studies in quantum physics. The textbook covers topics that are central to quantum physics: non-relativistic quantum mechanics, quantum statistical mechanics, relativistic quantum mechanics and quantum field theory. There is also background material on analysis, classical mechanics, relativity and probability. Each topic is explored through a statement of basic principles followed by simple examples. Around 100 problems throughout the textbook help readers develop their understanding.
Categories: Science

Topics in Quantum Mechanics

Topics in Quantum Mechanics

3.9 Some Brief Remarks on Axioms In the reference [ 84 ] , J. von Neumann
presents a rigorous mathematical foundation of quantum mechanics based on
the spectral theory of Hermitian ( = selfadjoint ) operators on a separable Hilbert
space .

Author: Floyd Williams

Publisher: Springer Science & Business Media

ISBN: 0817643117

Category: Mathematics

Page: 398

View: 217

This self-contained text presents quantum mechanics from the point of view of some computational examples with a mixture of mathematical clarity often not found in texts offering only a purely physical point of view. Emphasis is placed on the systematic application of the Nikiforov-- Uvarov theory of generalized hypergeometric differential equations to solve the Schr"dinger equation and to obtain the quantization of energies from a single unified point of view.
Categories: Mathematics

Quantum Reprogramming

Quantum Reprogramming

Ensembles and Single Systems: A Two-Tier Approach to Quantum Mechanics
E.J. Post ... In histreatise on the mathematical foundationsof quantum mechanics,
Johann von Neumann 3 established thatthe spectral theory ofHilbert spaces ...

Author: E.J. Post

Publisher: Springer Science & Business Media

ISBN: 9789401584104

Category: Philosophy

Page: 322

View: 833

Many, perhaps most textbooks of quantum mechanics present a Copenhagen, single system angle; fewer present the subject matter as an instrument for treating ensembles, but the two methods have been silently coexisting since the mid-Thirties. This lingering dichotomy of purpose for a major physical discipline has much shrouded further insights into the foundations of quantum theory. Quantum Reprogramming resolves this long-standing dichotomy by examining the mutual relation between single systems and ensembles, assigning each its own tools for treating the subject at hand: i.e., Schrödinger-Dirac methods for ensembles versus period integrals for single systems. A unified treatment of integer and fractional quantum Hall effects and a finite description of the electron's anomalies are mentioned as measures of justification for the chosen procedure of resolving an old-time dichotomy. The methods of presentation are, in part, elementary, with repetitive references needed to delineate differences with respect to standard methods. The parts on period integrals are developed with a perspective on elementary methods in physics, thus leading up to some standard results of de Rham theory and algebraic topology. Audience: Students of physics, mathematics, philosophers as well as outsiders with a general interest in the conceptual development of physics will find useful reading in these pages, which will stimulate further inquiry and study.
Categories: Philosophy

Mathematical Physics of Quantum Mechanics

Mathematical Physics of Quantum Mechanics

Quantitative analysis of metastability in reversible diffusion processes via a
Witten complex approach. Proceedings of the Symposium on Scattering and
Spectral Theory. Proceedings of the Symposium on Scattering and Spectral
Theory.

Author: Joachim Asch

Publisher: Springer

ISBN: 9783540342731

Category: Science

Page: 462

View: 357

This selection of outstanding articles – an outgrowth of the QMath9 meeting for young scientists – covers new techniques and recent results on spectral theory, statistical mechanics, Bose-Einstein condensation, random operators, magnetic Schrödinger operators and more. The book’s pedagogical style makes it a useful introduction to the research literature for postgraduate students. For more expert researchers it will serve as a concise source of modern reference.
Categories: Science

Self adjoint Extensions in Quantum Mechanics

Self adjoint Extensions in Quantum Mechanics

General Theory and Applications to Schrödinger and Dirac Equations with
Singular Potentials D.M. Gitman, I.V. Tyutin, B.L. Voronov ... The second part is
solving the spectral problem, i.e., a spectral analysis of the obtained observable.

Author: D.M. Gitman

Publisher: Springer Science & Business Media

ISBN: 9780817646622

Category: Science

Page: 511

View: 749

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.
Categories: Science

Spectral Theory Function Spaces and Inequalities

Spectral Theory  Function Spaces and Inequalities

... branches of mathematical analysis which include the spectral theory of both
ordinary and partial differential equations, ... a significant portion having been
concerned with problems arising in the study of non-relativistic quantum
mechanics.

Author: B. Malcolm Brown

Publisher: Springer Science & Business Media

ISBN: 9783034802635

Category: Mathematics

Page: 264

View: 648

This is a collection of contributed papers which focus on recent results in areas of differential equations, function spaces, operator theory and interpolation theory. In particular, it covers current work on measures of non-compactness and real interpolation, sharp Hardy-Littlewood-Sobolev inequalites, the HELP inequality, error estimates and spectral theory of elliptic operators, pseudo differential operators with discontinuous symbols, variable exponent spaces and entropy numbers. These papers contribute to areas of analysis which have been and continue to be heavily influenced by the leading British analysts David Edmunds and Des Evans. This book marks their respective 80th and 70th birthdays.
Categories: Mathematics