ADVANCES IN QUANTUM MECHANICS Edited by Paul Bracken Advances in Quantum Mechanics http://dx.doi.org/10.5772/50232 Edited by Paul Bracken Contributors Tokuzo Shimada, Gabino Torres-Vega, Francisco Bulnes, Inge S. Helland, Rodolfo Esquivel, Nelson Flores-Gallegos, Stephen Fulling, Fernando Mera, Jan Jerzy Slawianowski, Vasyl Kovalchuk, Fujii, Argyris Nicolaidis, Rafael De Lima Rodrigues, Constancio Miguel Arizmendi, Omar Gustavo Zabaleta, Peter Enders, GianCarlo Ghirardi, Donald Jack Kouri, Cynthia Whitney, Francisco De Zela, Douglas Singleton, Seyed Mohammad Motevalli, Yasuteru Shigeta, Valeriy Sbitnev, Jonathan Bentwich, Miloš Vaclav Lokajíček, John Ralston, L. M. Arevalo Aguilar, Carlos Robledo Sanchez, Paulo Cesar Garcia Quijas, Balmakov, Maricel Agop, Bjorn Jensen, Sergio Curilef, Flavia Pennini, Paul Bracken Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2013 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. However, users who aim to disseminate and distribute copies of this book as a whole must not seek monetary compensation for such service (excluded InTech representatives and agreed collaborations). 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. 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 Danijela Duric Technical Editor InTech DTP team Cover InTech Design team First published April, 2013 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechopen.com Advances in Quantum Mechanics, Edited by Paul Bracken p. cm. ISBN 978-953-51-1089-7 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface IX Section 1 The Classical-Quantum Correspondence 1 Chapter 1 Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables 3 Gabino Torres-Vega Chapter 2 Classical and Quantum Correspondence in Anisotropic Kepler Problem 23 Keita Sumiya, Hisakazu Uchiyama, Kazuhiro Kubo and Tokuzo Shimada Chapter 3 Charathéodory’s “Royal Road” to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics 41 Francisco De Zela Chapter 4 The Improvement of the Heisenberg Uncertainty Principle 67 L. M. Arévalo Aguilar, C. P. García Quijas and Carlos Robledo- Sanchez Section 2 The Schrödinger Equation 79 Chapter 5 Schrödinger Equation as a Hamiltonian System, Essential Nonlinearity, Dynamical Scalar Product and some Ideas of Decoherence 81 Jan J. Sławianowski and Vasyl Kovalchuk Chapter 6 Schrödinger Equation and (Future) Quantum Physics 105 Miloš V. Lokajíček, Vojtěch Kundrát and Jiří Procházka Chapter 7 Quantum Damped Harmonic Oscillator 133 Kazuyuki Fujii Section 3 Path Integrals 157 Chapter 8 The Schwinger Action Principle and Its Applications to Quantum Mechanics 159 Paul Bracken Chapter 9 Generalized Path Integral Technique: Nanoparticles Incident on a Slit Grating, Matter Wave Interference 183 Valeriy I. Sbitnev Chapter 10 Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals and Their Integral Transforms 213 Francisco Bulnes Section 4 Perturbation Theory 245 Chapter 11 Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces 247 Fernando D. Mera and Stephen A. Fulling Chapter 12 Quantum Perturbation Theory in Fluid Mixtures 269 S. M. Motevalli and M. Azimi Chapter 13 Quantal Cumulant Mechanics as Extended Ehrenfest Theorem 293 Yasuteru Shigeta Chapter 14 Unruh Radiation via WKB Method 317 Douglas A. Singleton Section 5 Foundations of Quantum Mechanics 333 Chapter 15 A Basis for Statistical Theory and Quantum Theory 335 Inge S. Helland Chapter 16 Relational Quantum Mechanics 361 A. Nicolaidis ContentsVI Chapter 17 On the Dual Concepts of 'Quantum State' and 'Quantum Process' 371 Cynthia Kolb Whitney Chapter 18 The Computational Unified Field Theory (CUFT): A Candidate 'Theory of Everything' 395 Jonathan Bentwich Chapter 19 Emergent un-Quantum Mechanics 437 John P. Ralston Chapter 20 The Wigner-Heisenberg Algebra in Quantum Mechanics 477 Rafael de Lima Rodrigues Chapter 21 New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations 499 Chia-Chun Chou, Mason T. Biamonte, Bernhard G. Bodmann and Donald J. Kouri Section 6 Quantization and Entanglement 519 Chapter 22 Quantum Dating Market 521 C. M. Arizmendi and O. G. Zabaleta Chapter 23 Quantization as Selection Rather than Eigenvalue Problem 543 Peter Enders Chapter 24 Entanglement, Nonlocality, Superluminal Signaling and Cloning 565 GianCarlo Ghirardi Chapter 25 The Husimi Distribution: Development and Applications 595 Sergio Curilef and Flavia Pennini Section 7 Quantum Information and Related Topics 621 Chapter 26 The Quantum Mechanics Aspect of Structural Transformations in Nanosystems 623 M. D. Bal’makov Contents VII Chapter 27 Decoding the Building Blocks of Life from the Perspective of Quantum Information 641 Rodolfo O. Esquivel, Moyocoyani Molina-Espíritu, Frank Salas, Catalina Soriano, Carolina Barrientos, Jesús S. Dehesa and José A. Dobado Chapter 28 The Theoretical Ramifications of the Computational Unified Field Theory 671 Jonathan Bentwich Chapter 29 Shannon Informational Entropies and Chemical Reactivity 683 Nelson Flores-Gallegos Chapter 30 A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics 707 Bjørn Jensen Chapter 31 Quantum Effects Through a Fractal Theory of Motion 723 M. Agop, C.Gh. Buzea, S. Bacaita, A. Stroe and M. Popa ContentsVIII Preface It can be stated that one of the greatest creations of twentieth century physics has been quan‐ tum mechanics. This has brought with it a revolutionary view of the physical world in its wake initiated by the work of people like Bohr, Schrödinger, Heisenberg and Born, Pauli and Dirac and many others. The development of quantum mechanics has taken physics in a vastly new direction from that of classical physics from the very start. This is clear from the compli‐ cated mathematical formalism of quantum mechanics and the intrinsic statistical nature of measurement theory. In fact, there continue at present to be many developments in the subject of a very fundamental nature, such as implications for the foundations of physics, physics of entanglement, geometric phases, gravity and cosmology and elementary particles as well. Quantum mechanics has had a great impact on technology and in applications to other fields such as chemistry and biology. The intention of the papers in this volume is to give research‐ ers in quantum mechanics, mathematical physics and mathematics an overview and introduc‐ tion to some of the topics which are of current interest in this area. Of the 29 chapters, the range of topics to be presented is limited to discussions on the founda‐ tions of quantum mechanics, the Schrödinger equation and quantum physics, the relationship of the classical-quantum correspondence, the impact of the path integral concept on quantum mechanics, perturbation theory, quantization and finally some informational-entropy aspects and application to biophysics. Many of the papers could be placed into more than one of these sections, so their breadth is quite substantial. The book has been put together by a large international group of invited authors and it is neces‐ sary to thank them for their hard work and contributions to the book. I gratefully acknowledge with thanks to the assistance provided by Ms. Danijela Duric who was publishing manager dur‐ ing the publishing process, and Intech publishing group for the publication of the book. Professor Paul Bracken Department of Mathematics, University of Texas, Edinburg, TX USA [...]... discuss quantum scars using energy values in (3) in order to facilitate comparison with literature 2.1.2 Matrix diagonalization in Sturmian basis We here summarize WMB method for efficient matrix diagonalization Firstly, in the Sturmian basis 26 4 Advances in Quantum Mechanics Advances in Quantum Mechanics 1 r �� |nℓm� = r � λr n! e− 2 (λr )ℓ+1 L2ℓ+1 (λr )Yℓm (θ, ϕ) n (2ℓ + n + 1) ! (4) with a scaling parameter... same object, with f the classical analogue of the spectrum of a quantum operator Continuing in a similar way, we can obtain the relationships shown in the following diagram Diagram 1 11 12 Advances in Quantum Mechanics where the constant s has units of action, length times momentum, the same units as the quantum constant ℏ Some of the things to note are: The operator e g L F is the eigenoperator of... which can be used as coordinates for representing dynamical quantities Another benefit of knowing the influence of conjugate dynamical variables on themselves and of using the same language for both theories lies in that some puzzling things that are found in one of the theories can be analysed in the other and this helps in the understanding of the original puzzle This is the case of the Pauli theorem... the plane and we can distinguish between the regions of phase space with negative or positive momen‐ tum One is to use half lines and t in the range from -π to π, with the curve t = 0 coinciding with the positive p axes The other option is to use the complete curve including positive 17 18 Advances in Quantum Mechanics and negative momentum values and with t ∈ ( - π / 2, π / 2) In the first option, the... Concluding remarks Once that we have made use of the same concepts in both classical and quantum mechanics, it is more easy to understand quantum theory since many objects then are present in both theories Actually, there are many things in common for both classical and quantum systems, as is the case of the eigensurfaces and the eigenfunctions of conjugate variables, which can be used as coordinates... densities [1] and other define functions of two variables from single variable quantum wave functions [2,3] Our approach is to use the same concepts in both types of dynamics but in their own realms, not using foreign unnatural objects In this chapter, we derive many inter relationships be‐ tween conjugate variables 1.1 Conjugate variables An important object in Quantum Mechanics is the eigenfunctions... 24 2 Advances in Quantum Mechanics Advances in Quantum Mechanics which indicates remnants of tori (cantori) in the classical phase space [9] Thus, over two decades from the early 70th, AKP was a good testing ground of theories (along with billiards) as well as a constant source of important information to quantum chaos studies However, there has not been much recent theory investigation on AKP Especially,... The localization patterns in the wave functions or Husimi functions are swapped between two eigenstates of energy at every avoiding crossing Repeating successively this swap process characteristic scarring patterns follow the POs responsible to them In this sense the quantum scarring phenomena are robust We conclude in section 4 2 Manifestation of Scars in AKP We first explain how we have prepared the... the chain rule, we have that {t, H } = ∑ i ( ∂t ∂H ∂ q i ∂ pi - ∂H ∂t ∂ q i ∂ pi )=∑ ( i dt d q i d q i dt + d pi dt dt d pi )= dt dt =1 (2) Now, a point in cotangent space can be specified as the intersection of 2n hypersurfaces A set of 2n independent, intersecting, hypersurfaces can be seen as a coordinate system in co‐ tangent space, as is the case for the hyper surfaces obtained by fixing values...Section 1 The Classical -Quantum Correspondence Chapter 1 Classical and Quantum Conjugate Dynamics – The Interplay Between Conjugate Variables Gabino Torres-Vega Additional information is available at the end of the chapter http://dx.doi.org/10.5772/53598 1 Introduction There are many proposals for writing Classical and Quantum Mechanics in the same lan‐ guage Some approaches use . ADVANCES IN QUANTUM MECHANICS Edited by Paul Bracken Advances in Quantum Mechanics http://dx.doi.org/10.5772/50232 Edited. Nanoparticles Incident on a Slit Grating, Matter Wave Interference 183 Valeriy I. Sbitnev Chapter 10 Quantum Intentionality and Determination of Realities in the Space-Time