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Wave Packets can Factorize Numbers

Holger Mack, Marc Bienert, Florian Haug, Matthias Freyberger, Wolfgang P. Schleich

Published Jan 1, 2002• 20 citations
OpenAlex

Abstract

We draw attention to various aspects of number theory emerging in the time evolution of elementary quantum systems with quadratic phases. Such model systems can be realized in actual experiments. Our analysis paves the way to a new, promising and effective method to factorize numbers.

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The study of wave packet is of great significance in quantum mechanics, optics and fluid mechanics. However, in order to solve the strict evolution behavior of wave packet, it is necessary not only to determine the parameters of various physical quantities such as mass, but also to carry out the complex integration process in configuration space and momentum space. In the final analysis, the evolution behavior of wave packet is the evolution behavior of distribution function parameters with time. Using the method of grey system theory, the time-varying parameters are replaced by time-varying response series, and then the formal structure is constructed according to the physical meaning of time evolution to realize the approximate simulation and approximate calculation of wave function expression. The advantage of this method is that it can simulate the evolution of wave packet under the uncertainty of each physical quantity. Based on the grey system theory, this paper uses GM (1,1) model to replace the time response function of wave packet distribution, obtains the simulated evolution behavior of wave packet wave function by means of mathematical techniques such as variational method, and makes error analysis and correction. In practical application, the approximate evolution behavior of wave function can be obtained only by determining the parameters of expansion coefficient of wave function. In this paper, the Gauss wave packet is taken as an example to discuss the method. The grey system theory of mathematical modeling is introduced into physical calculation for the first time, and an approximate calculation method which can be widely used in wave packet evolution calculation is given.