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Engineering Sciences

Application of the General Analytic Approach Used for Modeling Transients in Open Thermodynamic Systems to Numerical Evaluation of Some Basic Quantum and Astrophysical Parameters

I. Tkachenko, Yu. Miroshnichenko, Boris Drenchev, O. Tkachenko, Victoria Miroshnichenko

Published Feb 2, 2024• 1 citation
Publisher

Abstract

Based on the fundamental physical notions and the early developed general approach for analyzing a transient development in an open thermodynamic system, equivalence of the two formulations of the uncertainty principle in general physics was shown.Necessity of time quantum introduction proceeding from the uncertainty principle combined with the approach was grounded and an analytical formula to evaluate its numerical value was derived from the relation connecting uncertainties of energy and time.Using the approach, two ways for the obtained formula justification were proposed and numerically verified.Mandatory existence of the time quantum at beginning stages of the thermodynamic system evolution and its disappearing at the finishing ones was noted.Reciprocal dependence of the time quantum value on the energy of an open thermodynamic system is also emphasized that is in accordance with general physical notions.Coincidence was shown of the numerical data for some fundamental characteristics evaluated using the derived relations with * Corresponding author.

DOI
10.7546/engsci.lx.23.04.01

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Based on the fundamental physical notions and the early developed general approach for analyzing a transient development in an open thermodynamic system, equivalence of the two formulations of the uncertainty principle in general physics was shown.Necessity of time quantum introduction proceeding from the uncertainty principle combined with the approach was grounded and an analytical formula to evaluate its numerical value was derived from the relation connecting uncertainties of energy and time.Using the approach, two ways for the obtained formula justification were proposed and numerically verified.Mandatory existence of the time quantum at beginning stages of the thermodynamic system evolution and its disappearing at the finishing ones was noted.Reciprocal dependence of the time quantum value on the energy of an open thermodynamic system is also emphasized that is in accordance with general physical notions.Coincidence was shown of the numerical data for some fundamental characteristics evaluated using the derived relations with * Corresponding author.
Annotation. Based on the clear physically interpreted variant of the Heizenberg uncertainty relation connecting the uncertainties for impulse and a space coordinate of any material object, the other variant related to the uncertainties of time and energy for such an object was analytically derived based on the classical mechanics. Considering a measuring procedure as the combination of minimum two non-stationary stochastic processes a formula to define the time –quantum was derived using the early developed analytic model. Numerical value of the time – quantum then evaluated was shown to be in a good accordance with the known characteristics of our Universe. Dependence of the time-quantum value from the properties of material objects comprised and the evolution time for the Universe was shown.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
It is shown, that quantum theory with complex evolutionary time parameter and non-Hermitian Hamiltonian structure can be used for natural unification of quantum and thermodynamic principles. The theory is postulated as analytical in respect to the parameter of evolution, which real part is identified with the `usual' physical time, whereas the imaginary one is understood as proportional to the inverse absolute temperature. Also, the Hermitian part of the Hamiltonian is put equal to conventional operator of energy. It is shown, that the anti-Hermitian Hamiltonian part, which is taken as commuting with the energy operator, is constructed from parameters of decay of the system. It is established, that quantum dynamics, predicted by this theory, is integrable in the same sense as the corresponding non-modified one, and that it possesses a well defined arrow of time in isothermal and adiabatic regimes of the evolution. It is proved, that average value of the decay operator decreases monotonously (as the function of the physical time) in these important thermodynamical regimes for the arbitrary initial data taken. We discuss possible application of the general formalism developed to construction of time-irreversible modification of a string theory.
Abstract Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.
A precise form of the quantum-mechanical time-energy uncertainty relation is derived. For any given initial state (density operator), time-dependent Hamiltonian, and subspace of reference states, it gives upper and lower bounds for the probability of finding the system in a state in that subspace at a later or earlier time. The bounds involve only the initial data, the energy uncertainty in the initial state, and the energy uncertainty in the reference subspace. They describe how fast the state enters or leaves the reference subspace. They are exact if, but not only if, the initial state or the projection onto the reference subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, which generalizes the usual uncertainty relation. By introducing suitable comparison dynamics (trial propagators), the bounds can be made arbitrarily tight. They represent a time-dependent variational principle, in terms of trial propagators, which provides explicit error estimates and reproduces the exact time evolution when one varies over all trial propagators. As illustrations, we derive accurate lower bounds on the escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a He+{\mathrm{He}}^{+}He+ ion moving in a uniform magnetic field loses its electron.
We derive the uncertainty relation for a quantum open system consisting of a Brownian particle interacting with a bath of quantum oscillators at finite temperature. We examine how the quantum and thermal fluctuations of the environment contribute to the uncertainty in the canonical variables of the system. We show that upon contact with the bath (assumed to be ohmic in this paper) the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time in similar models in the discussion of quantum to classical transition. This offers some insight into the physical mechanisms involved in the environment-induced decoherence process. We obtain closed analytic expressions for this generalized uncertainty relation under the conditions of high temperature and weak damping, separately. We also consider under these conditions an arbitrarily squeezed initial state and show how the squeeze parameter enters in the generalized uncertainty relation. Using these results we examine the transition of the system from a quantum pure state to a nonequilibrium quantum statistical state and to an equilibrium quantum statistical state. The three stages are marked by the decoherence time and the relaxation time, respectively. With these observations we explicate the physical conditions under which the two basic postulates of quantum statistical mechanics become valid. We also comment on the inappropriate usage of the word “classicality” in many decoherence studies of quantum to classical transition.