One-Dimensional Three-Body Problem

Hurley, James A.

doi:10.1063/1.1705281

Cited by 15 publications

(4 citation statements)

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“…It is obvious to see that this potential is the sum of the harmonic oscillator and inversely quadratic potential. Hurley found that this kind of pseudoharmonic oscillator interaction between the particles can be exactly solved when he studied three-body problem in one dimension [198]. Since 1961 such a quantum system has been studied by many authors [2,[195][196][197][198][199][200][201][202][203][204][205][206][207][208][209].…”

Section: Introductionmentioning

confidence: 99%

Wave Equations in Higher Dimensions

Dong

2011

224

153

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No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. PrefaceThis work will introduce the wave equations in higher dimensions at an advanced level addressing students of physics, mathematics and chemistry. The aim is to put the mathematical and physical concepts and techniques like the wave equations, group theory, generalized hypervirial theorem, the Levinson theorem, exact and proper quantization rules related to the higher dimensions at the reader's disposal. For this purpose, we attempt to provide a comprehensive description of the wave equations including the non-relativistic Schrödinger equation, relativistic Dirac and Klein-Gordon equations in higher dimensions and their wide applications in quantum mechanics which complements the traditional coverage found in the existing quantum mechanics textbooks. Related to this field are the quantum mechanics and group theory. In fact, the author's driving force has been his desire to provide a comprehensive review volume that includes some new and significant results about the wave equations in higher dimensions drawn from the teaching and research experience of the author since the literature is inundated with scattered articles in this field and to pave the reader's way into this territory as rapidly as possible. We have made the effort to present the clear and understandable derivations and include the necessary mathematical steps so that the intelligent and diligent reader is able to follow the text with relative ease, in particular, when mathematically difficult material is presented. The author also embraces enthusiastically the potential of the LaTeX typesetting language to enrich the presentation of the formulas as to make the logical pattern behind the mathematics more transparent. In addition, any suggestions and criticism to improve the text are most welcome. It should be pointed out that the main effort to follow the text and master the material is left to the reader even though this book makes an effort to serve the reader as much as was possible for the author. This book starts out in Chap. 1 with a comprehensive review for the wave equations in higher dimensions and builds on this to introduce in Chap. 2 the fundamental theory about the SO(N ) group to be used in the successive Chaps. 3-5 including the non-relativistic Schrödinger equation, relativistic Dirac and Klein-Gordon equations. As important applications in non-relativistic quantum mechanics, from Chap. 6 to Chap. 12, we shall apply the theories proposed in Part II to study some vii viii Preface important quantum systems such as the harmonic oscillator, Coulomb potential, the Levinson theorem, generalized hypervirial theorem, exact and proper...

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Section: Introductionmentioning

confidence: 99%

Wave Equations in Higher Dimensions

Dong

2011

224

153

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“…Advantages of the pseudo-harmonic potential have been considered for improvements in the conventional presentation of molecular vibrations75. Hurley found that this kind of PHO interaction between the particles can be exactly solved by the separation of variables while studying the three-body problem in one dimension76. A few years later, Calogero studied the one-dimensional three- and N-body problems interacting pairwise via harmonic and inverse square (centrifugal) potential7778.…”

Section: Parity Deformed Jcm and Interpretationmentioning

confidence: 99%

Parity Deformed Jaynes-Cummings Model: “Robust Maximally Entangled States”

Dehghani¹,

Mojaveri²,

Shirin³

et al. 2016

Sci Rep

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The parity-deformations of the quantum harmonic oscillator are used to describe the generalized Jaynes-Cummings model based on the λ-analog of the Heisenberg algebra. The behavior is interestingly that of a coupled system comprising a two-level atom and a cavity field assisted by a continuous external classical field. The dynamical characters of the system is explored under the influence of the external field. In particular, we analytically study the generation of robust and maximally entangled states formed by a two-level atom trapped in a lossy cavity interacting with an external centrifugal field. We investigate the influence of deformation and detuning parameters on the degree of the quantum entanglement and the atomic population inversion. Under the condition of a linear interaction controlled by an external field, the maximally entangled states may emerge periodically along with time evolution. In the dissipation regime, the entanglement of the parity deformed JCM are preserved more with the increase of the deformation parameter, i.e. the stronger external field induces better degree of entanglement.

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“…Por exemplo, Landau e Lifshitz determinaram as soluções exatas da equação de Schrödinger para uma partícula em um potencial pseudo-harmônico em três dimensões [3]. Hurley considerou este potencial para estudar o problema de três corpos em uma dimensão [4]. Calogero abordou o problema de N corpos interagindo através de um potencial pseudo-harmônico [5,6].…”

Section: Introductionunclassified