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Journal of Modern Physics

Circular Scale of Time and Energy of a Quantum State Calculated from the Schrödinger Perturbation Theory

Stanisław Olszewski

Published Jan 1, 2014• 5 citations
Publisher

Abstract

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.

DOI
10.4236/jmp.2014.515152

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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 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.
An essential simplification of approach to the Schrödinger perturbation series for energy does hold when the perturbation events are arranged along a circular scale of time. The aim of the present paper is to demonstrate how such a scale of time leads to the recurrence calculation process of the Schrödinger energy terms belonging to an arbitrary perturbation order N. This process seems to have never been represented before. Only a non-degenerate quantum state and its perturbation due to the space-dependent potential are considered in the paper.
We point out that a suitable scale of time for the Schrödinger perturbation process is a closed line having rather a circular and not a conventional straight-linear character. A circular nature of the scale concerns especially the time associated with a particular order N of the perturbation energy which provides us with a full number of the perturbation terms predicted by Huby and Tong. On the other hand, a change of the order N—connected with an increased number of the special time points considered on the scale—requires a progressive character of time. A classification of the perturbation terms is done with the aid of the time-point contractions present on a scale characteristic for each N. This selection of terms can be simplified by a partition procedure of the integer numbers representing N-1. The detailed calculations are performed for the perturbation energy of orders N=7 and N=8 .
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.