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Bulletin of the Crimean Astrophysical Observatory

Вероятностные модели в классической и квантовой механике: Исправления к опубликованной ранее статье

V.Yu. Terebizh, В.Ю. Теребиж

Published Jun 1, 2011• 0 citations
OpenAlex

Abstract

Characteristic features of the stochastic models used in the classical mechanics, statistical physics and quantum theory are discussed. The arguments are given in favor of point of view that Nature, in itself, does not include probabilities. Researcher is forced to introduce the probabilistic concepts and corresponding models in conditions providing only partial predictability of a studied system behavior. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.

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Characteristic features of the stochastic models used in the classical mechanics, statistical physics and quantum theory are discussed. The arguments are given in favor of point of view that Nature, in itself, does not include probabilities. Researcher is forced to introduce the probabilistic concepts and corresponding models in conditions providing only partial predictability of a studied system behavior. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.
Characteristic features of the probabilistic models used in classical mechanics, statistical physics, and quantum theory are discussed. The viewpoint according to which there is no room for probabilities in Nature, as such, is consistently substantiated; the researcher is forced to introduce probabilistic concepts and the corresponding models in conditions that provide only partial predictability of the phenomena being studied. This approach allows one to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the theory of deterministic chaos, Boltzmann’s H -theorem, and paradoxes of quantum mechanics.
Quantum mechanics based on the probabilistic approach provides a powerful tool for accurate prediction and interpretation of quantum phenomena, allowing statistically sound predictions about the behavior of microparticles and quantum systems. This statement emphasizes the probabilistic nature of quantum mechanics, its applicability to quantum phenomena and microparticles, as well as the statistical nature of its predictions when applied to the macro effects of classical physics. In addition, the role of statistics and probability in various fields of science, such as particle physics, thermodynamics, biology, sociology, psychology, economics and finance, is discussed. The philosophical implications of the probabilistic approach and the associated limitations and challenges are also considered.
Statistical classical mechanics and quantum mechanics are developed and well-known theories that represent a basis for modern physics. Statistical classical mechanics enable the derivation of the properties of large bodies by investigating the movements of small atoms and molecules which comprise these bodies using Newton's classical laws. Quantum mechanics defines the laws of movement of small particles at small atomic distances by considering them as probability waves. The laws of quantum mechanics are described by the Schrödinger equation. The laws of such movements are significantly different from the laws of movement of large bodies, such as planets or stones. The two described theories are well known and have been well studied. As these theories contain numerous paradoxes, many scientists doubt their internal consistencies. However, these paradoxes can be resolved within the framework of the existing physics without the introduction of new laws. To clarify the paper for the inexperienced reader, we include certain necessary basic concepts of statistical physics and quantum mechanics in this paper without the use of formulas. Exact formulas and explanations are included in the Appendices. The text is supplemented by illustrations to enhance the understanding of the paper. The paradoxes underlying thermodynamics and quantum mechanics are also discussed. The approaches to the solutions of these paradoxes are suggested. The first approach is dependent on the influence of the external observer (environment), which disrupts the correlations in the system. The second approach is based on the limits of the self-knowledge of the system for the case in which both the external observer and the environment are included in the considered system. The concepts of observable dynamics, ideal dynamics, and unpredictable dynamics are introduced. The phenomenon of complex (living) systems is contemplated from the point of view of these dynamics.
This article analyzes the fundamental differences between classical and quantum mechanics from a scientific and theoretical perspective. By comparing the historical development of two physical paradigms, their mathematical frameworks, laws, and areas of application, the study explores their role in explaining natural phenomena. Classical mechanics is based on causal determinism, continuity, and the precise trajectory of motion, whereas quantum mechanics employs concepts of probability, discreteness, and the wave function. To highlight the differences between the theories, key physical laws such as Heisenberg’s uncertainty principle, the Schrödinger equation, and Newton’s laws are compared. The article also discusses the modern interpretation of mechanics in the era of quantum technologies.
Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.