THEORETICAL CONCEPTS OF QUANTUM MECHANICS Edited by Mohammad Reza Pahlavani Theoretical Concepts of Quantum Mechanics Edited by Mohammad Reza Pahlavani Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Maja Bozicevic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Theoretical Concepts of Quantum Mechanics, Edited by Mohammad Reza Pahlavani p. cm. ISBN 978-953-51-0088-1 Contents Preface IX Chapter 1 Complementarity in Quantum Mechanics and Classical Statistical Mechanics 1 Luisberis Velazquez Abad and Sergio Curilef Huichalaf Chapter 2 The Physical Nature of Wave/Particle Duality 23 Marcello Cini Chapter 3 The Bicomplex Heisenberg Uncertainty Principle 39 Raphaël Gervais Lavoie and Dominic Rochon Chapter 4 Correspondence, Time, Energy, Uncertainty, Tunnelling, and Collapse of Probability Densities 65 Gabino Torres–Vega Chapter 5 Anisotropic Kepler Problem and Critical Level Statistics 81 Kazuhiro Kubo and Tokuzo Shimada Chapter 6 Theory of Elementary Particles Based on Newtonian Mechanics 107 Nikolai A. Magnitskii Chapter 7 Better Unification for Physics in General Through Quantum Mechanics in Particular 127 Cynthia Kolb Whitney Chapter 8 Nonrelativistic Quantum Mechanics with Fundamental Environment 161 Ashot S. Gevorkyan Chapter 9 Non Commutative Quantum Mechanics in Time - Dependent Backgrounds 187 Antony Streklas VI Contents Chapter 10 Quantum Mechanics and Statistical Description of Results of Measurement 205 Lubomír Skála and Vojtěch Kapsa Chapter 11 Application of the Nikiforov-Uvarov Method in Quantum Mechanics 225 Cüneyt Berkdemir Chapter 12 Solutions for Time-Dependent Schrödinger Equations with Applications to Quantum Dots 253 Ricardo J. Cordero-Soto Chapter 13 The Group Theory and Non-Euclidean Superposition Principle in Quantum Mechanics 263 Nicolay V. Lunin Chapter 14 The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects 289 Francisco De Zela Chapter 15 Bohmian Trajectories and the Path Integral Paradigm – Complexified Lagrangian Mechanics 313 Valery I. Sbitnev Chapter 16 A Fully Quantum Model of Big Bang 341 S. P. Maydanyuk, A. Del Popolo, V. S. Olkhovsky and E. Recami Chapter 17 Spontaneous Supersymmetry Breaking, Localization and Nicolai Mapping in Matrix Models 383 Fumihiko Sugino Chapter 18 Correspondences of Scale Relativity Theory with Quantum Mechanics 409 Călin Gh. Buzea, Maricel Agop and Carmen Nejneru Chapter 19 Approximate Solutions of the Dirac Equation for the Rosen-Morse Potential in the Presence of the Spin-Orbit and Pseudo-Orbit Centrifugal Terms 445 Kayode John Oyewumi Chapter 20 Quantum Mechanics Entropy and a Quantum Version of the H-Theorem 469 Paul Bracken Chapter 21 Correction, Alignment, Restoration and Re-Composition of Quantum Mechanical Fields of Particles by Path Integrals and Their Applications 489 Francisco Bulnes Contents VII Chapter 22 The ‘Computational Unified Field Theory’ (CUFT): Harmonizing Quantum and Relativistic Models and Beyond 515 Jonathan Bentwich Chapter 23 Theoretical Validation of the Computational Unified Field Theory (CUFT) 551 Jonathan Bentwich Preface Classical physics breaks down to the level of atoms and molecules. This was made possible by the invention of a new apparatus that enabled the introduction of measurements in microscopic area of physics. There were two revolutions in the way we viewed the physical world in the twentieth century: relativity and quantum mechanics. Quantum mechanics was born in 1924, through the work of Einstein, Rutherford and Bohr, Schrödinger and Heisenberg, Born, Dirac, and many others. The principles of quantum mechanics that were discovered then are the same as we know them today. They have become the framework for thinking about most of the phenomena that physicists study, from simple systems like atoms, molecules, and nuclei to more exotic ones, like neutron stars, superfluids, and elementary particles. It is well established today that quantum mechanics, like other theories, has two aspects: the mathematical and conceptual. In the first aspect, it is a consistent and elegant theory and has been immensely successful in explaining and predicting a large number of atomic and subatomic phenomena. But in the second one, it has been a subject of endless discussions without agreed conclusions. Actually, without quantum mechanics, it was impossible to understand the enormous phenomena in microscopic physics, which does not appear in our macroscopic world. In this endless way of success for quantum mechanics, mathematics, especially mathematical physics developed to help quantum mechanics. It is believed that in order to be successful in theoretical physics, physicists should be professional mathematicians. Although this book does not cover all areas of theoretical quantum mechanics, it can be a reference for graduate students and researchers in the international community. It contains twenty tree chapters and the brief outline of the book is as follows: The first six chapters cover different aspects of the foundation of quantum mechanics, which is very important to understand quantum mechanics well. Chapters seven to twenty one discuss some mathematical techniques for solving the Schrodinger differential equation that usually appears in all quantum mechanical problems. Next two chapters of this volume are related to computational unified field theory, where the Schrodinger equation is not necessarily valid in its regular form. X Preface This book is written by an international group of invited authors and we would like to thank all of them for their contributions to this project. I gratefully acknowledge the assistance provided by Ms. Maja Bozicevic as the Publishing Process Manager during the publishing process, and InTech publishing team for the publication of this book. Mohammad Reza Pahlavani Head of Nuclear Physics Department, Mazandaran University, Mazandaran, Babolsar, Iran [...]... generalization of quantum hypothesis of light developed by Planck and Einstein for any kind of microparticles (14) The 4 4 Theoretical Concepts of Quantum Mechanics Will-be-set-by-IN-TECH experimental confirmation of these wave-particle duality for any kind of matter revealed the unity of material world In fact, wave-particle duality is a property of matter as universal as the fact that any kind of matter... analogous perspective as the finite character of the speed of light c implies the impossibility of a sharp separation between the notions of space and time, the finite character of the quantum of action h implies the impossibility of a sharp separation between the behavior of a quantum system ¯ and its interaction with the measuring instruments In the early days of quantum mechanics, Bohr understood that complementarity... generation of physicists the ambiguities which still remained unsolved, and stimulated a renewed interest on those conceptual foundations of the theory which had been set aside under the impact of the the extraordinary experimental and theoretical boom of physics triggered at the end of World War 2 by the opening of the Nuclear Era 24 Theoretical Concepts of Quantum Mechanics The central issue of the... regarded as the quantum of entropy Classical mechanics provides a precise description for the systems with large quantum numbers, or equivalently, in the limit h → 0 Similarly, thermodynamics appears as a suitable ¯ Complementarity in Quantum Mechanics and Classical Statistical Mechanics Complementarity in Quantum Mechanics and Classical Statistical Mechanics Comparison criterium Quantum mechanics parametrization... ensemble that corresponds to the so-called mixed quantum state, whose description is performed in terms of the ˆ density matrix ρ The consideration of the density matrix is the natural description of quantum statistical mechanics Complementarity in Quantum Mechanics and Classical Statistical Mechanics Complementarity in Quantum Mechanics and Classical Statistical Mechanics 5 5 Superposition principle is the... action of the system associated with the known Hamilton-Jacobi theory of classical mechanics Physically, this principle expresses that quantum mechanics contains classical mechanics as an asymptotic theory At the same time, it states that quantum mechanics should be formulated under the correspondence with classical mechanics Physically speaking, it is impossible to introduce a consistent quantum mechanics. .. of the question but is focused on the exposure of the results of more than twenty years of research of my group in Rome, which in my opinion provide a possible way of connecting together at the same time the random nature of the events at the atomic level of reality and the completeness of their probabilistic representation by the principles of Quantum Mechanics 1.2 The two slits experiment In order... role of measuring instruments Historically, correspondence principle was formally introduced by Bohr in 1920 (16), although he previously made use of it as early as 1913 in developing his model of the atom 6 6 Theoretical Concepts of Quantum Mechanics Will-be-set-by-IN-TECH (17) According to this principle, quantum description should be consistent with classical description in the limit of large quantum. .. measurements of the quantum system under the same initial conditions Abstractly, this procedure is equivalent to consider simultaneous measurements over a quantum statistical ensemble: such as an infinite set of identical copies of the quantum system, which have been previously prepared under the same experimental procedure2 Due to the important role of measurements in the knowledge state of quantum systems, quantum. .. Complementarity in Quantum Mechanics and Classical Statistical Mechanics Complementarity in Quantum Mechanics and Classical Statistical Mechanics 21 21 of complementarity has been focused in those systems in thermodynamic equilibrium, the operational interpretation discussed in this chapter strongly suggests the existence of a counterpart of Schrödinger equation (42) in classical statistical mechanics In . THEORETICAL CONCEPTS OF QUANTUM MECHANICS Edited by Mohammad Reza Pahlavani Theoretical Concepts of Quantum Mechanics Edited by Mohammad. performed in terms of the density matrix ˆ ρ. The consideration of the density matrix is the natural description of quantum statistical mechanics. 4 Theoretical Concepts of Quantum Mechanics Complementarity. that leads to expression (10). 6 Theoretical Concepts of Quantum Mechanics Complementarity in Quantum Mechanics and Classical Statistical Mechanics 7 2.2.4 Operators of physical observables and Schrödinger