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Studia Philosophiae Christianae

Kołowa skala czasu z punktu widzenia mechaniki kwantowej w ujęciu Schrödingera

Stanisław Olszewski

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

A basic approximate method of non-relativistic quantum mechanics is Schrodinger's perturbation theory. Two kinds of the time scale can be applied in order to follow the events which lead from an unperturbed non-degenerate stationary quantum state to a similar perturbed state. Topologically, one time scale is a straight-linear scale similar to that applied by Feynman, but the another possible scale is a topological circle. It is demonstrated that in calculating large orders of the perturbation energy a circular scale of time is much more efficient than a straight-linear scale.

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A basic approximate method of non-relativistic quantum mechanics is Schrodinger's perturbation theory. Two kinds of the time scale can be applied in order to follow the events which lead from an unperturbed non-degenerate stationary quantum state to a similar perturbed state. Topologically, one time scale is a straight-linear scale similar to that applied by Feynman, but the another possible scale is a topological circle. It is demonstrated that in calculating large orders of the perturbation energy a circular scale of time is much more efficient than a straight-linear scale.
The paper presents a circular scale of time—and its diagrams—which can be successfully applied in calculating the Schrödinger perturbation energy of a non-degenerate quantum state. This seems to be done in a more simple way than with the aid of any other of the perturbation approaches of a similar kind. As an example of the theory suitable to comparison is considered the Feynman diagrammatic method based on a straight-linear scale of time which represents a much more complicated formalism than the present one. All diagrams of the approach outlined in the paper can obtain as their counterparts the algebraic formulae which can be easily extended to an arbitrary Schrödinger perturbation order. The calculations and results descending from the perturbation orders N between N = 1 and N = 7 are reported in detail.
The main facts about the scale of time considered as a plot of a sequence of events are submitted both to a review and a more detailed calculation. Classical progressive character of the time variable, present in the everyday life and in the modern science, too, is compared with a circular-like kind of advancement of time. This second kind of the time behaviour can be found suitable when a perturbation process of a quantum-mechanical system is examined. In fact the paper demonstrates that the complicated high-order Schrodinger perturbation energy of a non-degenerate quantum state becomes easy to approach of the basis of a circular scale. For example for the perturbation order N = 20 instead of 19! ≈ 1.216 × 1017 Feynman diagrams, the contribution of which should be derived and calculated, only less than 218 ≈ 2.621 × 105 terms belonging to N = 20 should be taken into account to the same purpose.
The paper is a kind of a review which considers an investigation of the scale of time suggested by an application of the Schrödinger perturbation method, especially when the perturbation of a non-degenerate quantum state is examined. In fact the method was applied in numerous cases—also by Schrödinger himself—without any use of the notion of time. Simultaneously, because of the development of computers, their use in solving the perturbation problems gradually decreased. However, the point of importance in the paper became the time. We demonstrate that collisions of a quantum system with the perturbation potential can be arranged along a circular scale of time whose properties provide us precisely with the energy terms obtained by the Schrödinger perturbation theory. This validity of results is checked till the perturbation order N = 7.
The main differential equations of quantum theory are the eigenequations based on the energy operator; they have the energy as eigenvalues and the wave functions as eigenfunctions. A usual complexity of these equations makes their accurate solutions accessible easily only for very few physical cases. One of the methods giving the approximate solutions is the Schrödinger perturbation theory in which both the energies and wave functions of a more complicated eigenproblem are approached with the aid of similar parameters characteristic for a less complicated eigenproblem. No time parameter is necessary to be involved in these calculations. The present paper shows that the Schrödinger perturbation method for non-degenerate stationary quantum states, i.e. the states being independent of time, can be substantially simplified by applying a circular scale of time separately for each order of the perturbation theory. The arrangement of the time points on the scale, combined with the points contractions, gives almost immediately the series of terms necessary to express the stationary perturbation energy of a given eigenproblem. The Schrödinger’s method is compared with the Born-Heisenberg-Jordan perturbation approach.
Physically the examined perturbation problem can be regarded as a set of collision events of a time-independent perturbation potential with a quantum system. As an effect of collisions there is an expected definite change of energy of an initially unperturbed state of the system to some stationary perturbed state. The collision process certainly occupies some intervals of time which, however, do not enter the formalism. A striking property is the result of a choice of the sequence of collisions according to the applied circular scale of time: the scale produces almost automatically the energy terms predicted by the Schrödinger perturbation theory which usually is attained in virtue of complicated mathematical transformations. Beyond of the time scale and its rules—strictly connected with the perturbation order N introduced by Schrödinger—a partition process of the number N-1 is applied. This process, combined with contractions of the time points on the scale, provides us precisely with the perturbation terms entering the Schrödinger theory.