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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
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Notice
Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those
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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
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ISBN 978-953-51-1089-7
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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
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