Our work is divided into 6 parts. The first part is the chaos theory. An
introduction to a mathematical world and some of the biggest and most famous
experiments. Some of them are the attractors, the three-body problem and the
Principle of Uncertainty.
The second part is the Quantum Physics or Quantum Mechanics. Large
theories that changed the way we think and the way we perceive the universe as
many of the greatest physicists who have passed from this world, are referred to
this part.
The third part, and the shortest is the Quantum Chaos. A part of physics, that
studies how a quantum system behaves, whose classical limit is Chaotic. In
simpler words, the attempt to connect quantum physics and chaos. The fourth
part includes some biographical information of the greatest scientists. The fifth
and sixth part, are some explanations and our sources.
Our work is divided into 6 parts. The first part is the chaos theory. An
introduction to a mathematical world and some of the biggest and most famous
experiments. Some of them are the attractors, the three-body problem and the
Principle of Uncertainty.
The second part is the Quantum Physics or Quantum Mechanics. Large
theories that changed the way we think and the way we perceive the universe as
many of the greatest physicists who have passed from this world, are referred to
this part.
The third part, and the shortest is the Quantum Chaos. A part of physics, that
studies how a quantum system behaves, whose classical limit is Chaotic. In
simpler words, the attempt to connect quantum physics and chaos. The fourth
part includes some biographical information of the greatest scientists. The fifth
and sixth part, are some explanations and our sources.
This dissertation is divided into 5 parts. In the first part there is an extensive reference to the field of quantum mechanics and the way in which it was founded through concepts and axioms that characterize it, as well as an explanation of basic phenomena that could not be interpreted by classical physics. With the help of particle dualism and Dirac mathematical symbolism, we arrive at the physical interpretation of the wave function. With an extensive analysis of the basic quantum operators, we arrive at various ways of solving the Schrodinger equation.
The second part begins with classic automation systems and analysis of the basic classic portals through their simulation by two modern software, Simulink and logo soft comfort. The third part presents the quantum bit and the basic quantum gates and registers, as well as superposition states by analyzing various mathematical concepts such as Hilbert geometry, tensile product and internal product. The predominance of quantum computers is initially demonstrated by reference to a quantum game. The fourth part presents neural networks and computational intelligence. With the help of transfer functions we first explain the operation of the classical logic gates and later the quantum gates. The following is the construction of neural networks and the explanation of their operation with the help of commands of the modern Matlab software. In the fifth and last part of the work we simulate quantum circuits of 2 or 3 qubits with the help of 2 modern quantum simulators, Quirk Simulator and Qcs Karafyllidis as well as an online quantum computer Ibm q experience.
From the very beginning classical nonlinear dynamics has enjoyed much popularity even among the noneducated public as is documented by numerous articles in well-renowned magazines, including nonscientific papers. For its nonclassical counterpart, the quantum mechanics of chaotic systems, termed in short ‘quantum chaos’, the situation is completely different. It has always been considered as a more or less mysterious topic, reserved to a small exclusive circle of theoreticians. Whereas the applicability of classical nonlinear dynamics to daily life is comprehensible for a complete outsider, quantum chaos, on the other hand, seems to be of no practical relevance at all. Moreover, in classical nonlinear dynamics the theory is supported by numerous experiments, mainly in hydrodynamics and laser physics, whereas quantum chaos at first sight seems to be the exclusive domain of theoreticians. In the beginning the only experimental contributions came from nuclear physics [Por65]. This preponderance of theory seems to have suppressed any experimental effort for nearly two decades. The situation gradually changed in the middle of the eighties, since when numerous experiments have been performed. An introductory presentation also suited to the experimentalist with no or only little basic knowledge is still missing. It is the intention of this monograph to demonstrate that there is no reason to be afraid of quantum chaos. The underlying ideas are very simple. It is essentially the mathematical apparatus that makes things difficult and often tends to obscure the physical background.
The present work consists of eight chapters.
The first chapter gives a brief historical overview of the impasses that Classical Physics found in the late 19th and early 20th centuries, and the efforts made in the framework of Old Quantum theory to overcome it, ultimately leading to the foundation of Quantum Engineering in the 1920s.
The second chapter shows the ratio of the Schrodinger equation to the classical wave differential equation, the concept of wave function, the physical importance of probability density, continuity equation and probability current density, while indicating the conditions to be met by the wave function.
The third chapter examines square dynamic wells as well as the rectangular dynamic barrier, through the study of which the tunnel phenomenon emerges. The reflecting and transit coefficients are also calculated, while the tunnel effect is also being examined in relation to the principle of uncertainty.
The fourth chapter presents semi-classical approaches to the solution of the Schrodinger equation, with an emphasis on the WKB approach. The power of the WKB approach is analysed, and applied to a dynamic well, as well as in the case of tunneling through a dynamic barrier, by calculating the transit coefficient.
The fifth chapter analyses the time delay in the tunnel phenomenon and examines the concept of minimum transit time.
The sixth chapter refers to natural systems where the tunnel phenomenon occurs, such as cold emission of electrons from metals, a-decay and nuclear fusion. Finally, due to the purely undulating nature of the phenomenon, cases of observation are mentioned macroscopicly, such as the case of total internal reflection.
The seventh chapter presents a lesson plan for the teaching of the phenomenon in Secondary Education, using information and communication technologies in the educational process, and finally the main conclusions presented resulting from the previous study.
By now the “chaos revolution” has reached nearly every branch of the natural sciences. In fact, chaos is everywhere. To name but a few examples, we talk about chaotic weather patterns, chaotic chemical reactions and the chaotic evolution of insect populations. Atomic and molecular physics are no exceptions. At first glance this is surprising since atoms and molecules are well described by the linear laws of quantum mechanics, while an essential ingredient of chaos is nonlinearity in the dynamic equations. Thus, chaos and atomic physics seem to have little to do with each other. But recently, atomic and molecular physicists have pushed the limits of their experiments to such high quantum numbers that it starts to make sense, in the spirit of Bohr's correspondence principle, to compare the results of atomic physics experiments with the predictions of classical mechanics, which, for the most part, show complexity and chaos. The most striking observation in recent years has been that quantum systems seem to “know” whether their classical counterparts display regular or chaotic motion. This fact can be understood intuitively on the basis of Feynman's version of quantum mechanics. In 1948 Feynman showed that quantum mechanics can be formulated on the basis of classical mechanics with the help of path integrals. Therefore it is expected that the quantum mechanics of an atom or molecule is profoundly influenced, but of course not completely determined, by the qualitative behaviour of its underlying classical mechanics.
The major purpose of this study is to consecrate specific approaches to some problems of quantum theory and quantum cosmology, in terms of decoherence histories formalism which is a very promising formalism for the canonical quantum gravity theories. The reason is to give a unified picture to these issues in order to be possible to compare them. Specifically, the first part contains two different approaches to the time of arrival in quantum mechanics, both of these use a histories formalism. The first is from Halliwell and Yearsly (2009) and the second from Anastopoulos and Saviddou (2012). By comparing them we deduce that both of them first gives a coarse-grain form of the Kijowski' s probability distribution. The second part concerns quantum cosmology. In this, we presented a decoherent histories approach to quantum cosmological probabilities, in which was used a complex potential, from Halliwell (2009). After that we present a histories approach to minisuperspace models by Anastopoulos and Savidou (2005). In this, minisuperspace models is written in terms of histories projector operator (HPO) formalism. The spectacular of this is that in that form general relativity satisfies both spacetime diffeomorfisms and Dirac algebra, which is very important because it is easier to be quantized.