International Journal of Economics Business and Social Science Research
QUANTITATIVE THEORETICAL EVALUATION OF THE TIME QUANTUM VALUE BASED ON THE HEIZENBERG UNCERTAINTY PRINCIPLE AND THE ANALYTIC MODEL OF NONSTATIONARY STOCHASTIC PROCESSES
Igor Tkachenko, Serhii Miroshnychenko, Kostiantin Tkachenko, Vitaliy Miroshnychenko, Yurii Miroshnychenko
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.
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.
The physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time-energy relation is much more difficult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schro¨dinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schro¨dinger’s equation.
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.
The laws of classical mechanics and quantum theory do not distinguish between the past and the future. But a lot of phenomena tell us that the "arrow of time" points only in one direction. For example, in biology we discover evolutionary processes leading to diversification and increasing complexity; it is a typical feature of such evolutionary processes that "forward" in time is possible but "backward" in time is impossible. At what level can phenomena like these be described? There is another problem in connection with time: within usual quantum theory, space-coordinates and time are not symmetrical to each other, which is in contrast to the basic results of special theory of relativity. In usual quantum theory coordinates are operators and play the role of statistical quantities; time however does not behave statistically and remains a simple parameter when we go from classical to quantum mechanics. For this reason there is no uncertainty relation for the time and energy within usual quantum theory which would agree in its physical content with that for the coordinates and momenta. The purpose of this paper is to show, on the basis of new aspects, that both one-sidedness of time and its uncertainty are charateristics at the fundamental level of quantum theory. We have extended usual quantum theory on the basis of space-time arguments. In this connection a time-operator appeared and there is in general temporal asymmetry at the microscopic level. The picture might be important in the description of phenomena with an "arrow of time", i.e., processes which belong to the physics of becoming. Furthermore, we have derived an uncertainty relation for the energy and time which agrees in its physical content with the uncertainty relation for the momentum and the coordinate; this relation is different from that of usual quantum theory. Within the frame of this physical picture all the concepts (diversity, natural selection, mutation), which are presently used in connection with the evolution of biological systems, are already established at the quantum mechanical level and, therefore, our theoretical picture might also be the foundation for the explanation of the phenomena at the biological (macro-molecular) level.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.
It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3 have been derived, where ψ denotes the initial state, H the Hamiltonian, R a positive constant, and C is explicitly known. As intermediate step, bounds on the derivatives of the S-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3, and the constants Cn,K explicitly known.