The most likely value of a random variable is the value at which the probability has the largest value. The average value is a macroscopic parameter. A stochastic system can, of course, be described by the probabilities of the individual values of the random variable. The particular stochastic model discussed here is useful whenever the probing of a stochastic system alters the system. It is mainly used to formulate the quantum mechanical model of nature. Werner Heisenberg was the only major physicist to remain in Germany during World War II. He was in charge of the German nuclear weapons effort. In quantum mechanics, the motion of particles follows probabilistic rules. However, the probability itself propagates in conformity with the law of causality; that is, only the probability that a particle will have a certain position and will have a certain momentum can be calculated according to quantum mechanics.
The most likely value of a random variable is the value at which the probability has the largest value. The average value is a macroscopic parameter. A stochastic system can, of course, be described by the probabilities of the individual values of the random variable. The particular stochastic model discussed here is useful whenever the probing of a stochastic system alters the system. It is mainly used to formulate the quantum mechanical model of nature. Werner Heisenberg was the only major physicist to remain in Germany during World War II. He was in charge of the German nuclear weapons effort. In quantum mechanics, the motion of particles follows probabilistic rules. However, the probability itself propagates in conformity with the law of causality; that is, only the probability that a particle will have a certain position and will have a certain momentum can be calculated according to quantum mechanics.
Introduction to statistical ensembles The concept of a random or stochastic process or function represents a generalization of the idea of a set of random variables x 1 , x 2 , …, when the set is no longer countable and the variables form a continuum. We therefore introduce a continuous parameter t , such as time, that labels the variates. We call x ( t ) a random process or a random function of t if x does not depend on t in a deterministic way. Random processes are encountered in many fields of science, whenever fluctuations are present. Examples of a real random process x ( t ) are the fluctuating voltage across an electrical resistor, and the coordinates of a particle under-going Brownian motion. We shall see shortly that the optical field generated by any realistic light source must also be treated as a random function of position and time. Of course the parameter t may also stand for some quantity other than time, but for simplicity we shall take it to represent time. In our applications x ( t ) will frequently represent a Cartesian component of the electric or magnetic field vector in a light beam. To begin with we shall take x ( t ) to be real, but complex random processes will also be encountered. The ensemble average As x does not depend on t deterministically, we can only describe its values statistically, by some probability distribution or probability density.
Introduction Despite the fact that classical mechanics employs deterministic dynamical laws, random dynamical processes often arise in classical physics, as well as in everyday life. A stochastic or random process is one in which states-of-affairs at successive times are not related to one another by deterministic laws, and instead probability theory is employed to describe whatever regularities exist. Tossing a coin or rolling a die several times in succession are examples of stochastic processes in which the previous history is of very little help in predicting what will happen in the future. The motion of a baseball is an example of a stochastic process which is to some degree predictable using classical equations of motion that relate its acceleration to the total force acting upon it. However, a lack of information about its initial state (e.g., whether it is spinning), its precise shape, and the condition and motion of the air through which it moves limits the precision with which one can predict its trajectory. The Brownian motion of a small particle suspended in a fluid and subject to random bombardment by the surrounding molecules of fluid is a well-studied example of a stochastic process in classical physics. Whereas the instantaneous velocity of the particle is hard to predict, there is a probabilistic correlation between successive positions, which can be predicted using stochastic dynamics and checked by experimental measurements. In particular, given the particle's position at a time t , it is possible to compute the probability that it will have moved a certain distance by the time t + Δ t .
The source of inspiration for doing quantum probability lies in the ingenious methods adopted by physicists to compute probabilities of events concerning the subatomic world of elementary particles where the laws of classical mechanics break down and the distinction between a particle and a wave becomes vague. These methods lead to a generalisation of classical probability which may be described as a study of observable quantities concerning any system subject to chance and exhibiting the following basic features: (i) each observable quantity behaves like a classical real valued random variable on a probability space and hence has a probability distribution on the real line; (ii) it may be impossible to measure simultaneously two individually observable quantities in the sense that measurement of one may interfere with and destroy already existing information about the value of the other; (iii) bounded observable quantities constitute a real linear space in which the vectorial sum of two individually observable quantities interfering with each other is different from the arithmetic sum of their actual values. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
Introduction In our treatment of quantum mechanics we discussed the notion of the state of a given system. The state was specified by giving the eigenvalues of a maximal set of mutually commuting observables. This represented the most information we could have about a given system in principle. In practice, of course, we do not really have all of the necessary information about the eigenvalues available. No experimentalist sets out with the primary intention of obtaining results which correspond to a maximal set of mutually compatible observables for any macroscopic system, such as a gas. Even a very small volume of gas contains an enormous number of particles. The quantum mechanical state would require us to know a great deal of information about the internal states of the electrons and nuclei of the gas molecules. The experimentalist may only be concerned with gross properties such as the pressure, internal energy, or specific heat for example. Most of the detailed information about the detailed state of the system is neither observed, nor of direct interest. Instead we must content ourselves with a description of the average, or most likely, behaviour of the system in the probabilistic sense. The usual technique for doing this consists of the introduction of an ensemble, meaning a very large collection of systems whose measured macroscopic properties are the same as those for the system we are interested in, but whose unobserved properties can vary.
The fundamental question is this manuscript is: What are the fruits of our Scientific Journey during the last 300 years and do we realize that we always have a choice? That we always have the choice with our massive scientific powers to create Heaven on Earth or to create Hell on Earth. We always have the freedom of choice. But when we look at the fruits of our scientific powers it looks more like Hell on Earth than Heaven on Earth. How did we come so far?Did we really want, with our scientific powers, to create war machines that burn women and children alive in massive merciless wars like in Syria? Did we scientists work so hard and gave our lives the last 300 years to science to accomplish these final results? War and hunger everywhere. Massive Climate Changes that are coming. Massive areas where people live, will disappear under the Oceans. Waves of massive Hunger and Thirst are coming. Deserts of refugees who do not know anymore where to go. Forests will turn into deserts. Advance software will enslave the world and kill our privacy and our freedom. Was this really our choice?Or have we been misled? Mislead by a world we have forgotten. A world we do deny. A world that has tricked us by our own weakness and our own desires for acknowledgement and honor. Is there truth in the ancient story about the war between Light and Darkness? Was there truth in the famous words of Albert Einstein when he declared: ”God does not Play Dice”? An important role in the transformation of a fundamentally “Religious Newtonian Physics” into an Atheistic “Quantum Mechanical Physics” has been the contribution of Werner Heisenberg’s publication in Zeitschrift für Physik” in 1925 about the “Umdeutung” [42],[43],[44] and [45]. Heisenberg’s publication in 1925 was the final end of the religious fundamental principle of “Causality” in Physics and had been replaced by the “Probability Idolatry” represented by the “Heisenberg Uncertainty Principle” embodying the quantum mechanical probability waves as a fundamental solution of the quantum mechanical Schrödinger Wave Equation. Heisenberg’s "Umdeutung" paper transformed a physics based on a religious concept of "Causality" into an atheistic physics based on a atheistic concept of "Probablity". "Causality" is the foundation of every religion. What we do and how we act and respond has always consequences. Taking out this fundamental concept of “Causality” removes the religious origin of physics. Replacing “Causality” by the “Heisenberg Uncertainty Principle” turns the Creator of the Universe into a Dice and changes a world, in which human beings are valuable and their decisions determining our history and our future, into a meaningless universe in which humans are meaningless. Einstein could not accept that idea of a meaningless universe without a Creator. And Einstein’s famous equation E= m c2 became the foundation of the development of the “nuclear bomb” in which Werner Heisenberg chose for the Nazis and worked in a team to develop an atomic bomb for Adolph Hitler.Nazi Germany undertook several research programs relating to nuclear technology, including nuclear weapons and nuclear reactors, before and during World War II. These were variously called “Uranverein” (Uranium Club) or “Uranprojekt” (Uranium Project). When the second world war began, Heisenberg was made director of the German atom bomb project. He spent five years working on it. At war's end, Heisenberg was captured by the Allies and was imprisoned in England for six months. In the US Robert Oppenheimer became the director of the American Manhattan atom bomb project.