Preface

Chapter 27 **Decoding the Building Blocks of Life from the Perspective of**

Chapter 28 **The Theoretical Ramifications of the Computational Unified**

Chapter 30 **A Novel Isospectral Deformation Chain in Supersymmetric**

Chapter 31 **Quantum Effects Through a Fractal Theory of Motion 723** M. Agop, C.Gh. Buzea, S. Bacaita, A. Stroe and M. Popa

Chapter 29 **Shannon Informational Entropies and Chemical**

**Quantum Mechanics 707**

Rodolfo O. Esquivel, Moyocoyani Molina-Espíritu, Frank Salas, Catalina Soriano, Carolina Barrientos, Jesús S. Dehesa and José A.

**Quantum Information 641**

Dobado

**VIII** Contents

**Field Theory 671** Jonathan Bentwich

**Reactivity 683** Nelson Flores-Gallegos

Bjørn Jensen

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

**Section 1**

**The Classical-Quantum Correspondence**

**The Classical-Quantum Correspondence**

**Chapter 1**

*n*=0

*Q* with variable *z* =(*q*, *p*),

*<sup>∞</sup>* of a Hermi‐

**Classical and Quantum Conjugate Dynamics –**

There are many proposals for writing Classical and Quantum Mechanics in the same lan‐ guage. Some approaches use complex functions for classical probability 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‐

sentations, like the coordinate representation *ψn*(*q*)= *q* | *n* . The basis vector used to provide the coordinate representation, |*q* >, of the wave function are themselves eigenfunctions of

where *q* and *p* are *n* dimensional vectors representing the coordinate and momentum of point particles. We can associate to a dynamical variable *F* (*z*) its eigensurface, i.e. the level

. These eigenfunctions belong to a Hilbert space and can have several repre‐

^ We proceed to define the classical analogue of both objects, the

© 2013 Torres-Vega; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Torres-Vega; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

*<sup>Q</sup>* <sup>|</sup> *<sup>F</sup>* (*z*)= *<sup>f</sup>* } (1)

An important object in Quantum Mechanics is the eigenfunctions set {|*n* >}

Classical motion takes place on the associated cotangent space *T \**

<sup>Σ</sup>*<sup>F</sup>* ( *<sup>f</sup>* ) ={*<sup>z</sup>* <sup>∈</sup>*<sup>T</sup> \**

**The Interplay Between Conjugate Variables**

Additional information is available at the end of the chapter

Gabino Torres-Vega

**1. Introduction**

tween conjugate variables.

**1.1. Conjugate variables**

^

the coordinate operator *Q*

eigenfunction and its support.

tian operator *F*

set

http://dx.doi.org/10.5772/53598
