A quantum-like theory of wave propagation in periodic structures

Abstract

Recent investigations of wave propagation in numerical computing domains in which hyperbolic equations are approximated with finite differences (typically in the context of computational fluid dynamics) have brought to the fore striking similarities between this class of phenomena and corresponding aspects of quantum mechanics: those have been reported in Refs [l, 2]. This has pointed toward the possibility of formulating a theory that would parallel Schr6dinger's theory of particle physics, and that would explain those similarities. Further investigation did show that this was indeed the case: it was shown in Ref. [3] that, as expected (but nevertheless surprisingly), such a new formalism may indeed be established. While these results were obtained in the context of numerical computing, nothing in their mathematics was of such a nature that their applicability should remain restricted to that specific case. That this is indeed true is demonstrated in the present paper by extending those previous results to the description of wave propagation in the periodic system consisting of a taut, massless string to which point masses are attached almost periodically (that is to be considered as a model of a broad class of physical periodic systems). Other aspects of this may be found in Refs [4, 5]. Part of the new results consist in showing that many of the known properties of classical wave propagation in periodic structures may be described by a formalism and by mathematics similar to those of the wave mechanics component of quantum mechanics. They also show that certain phenomena that were considered as strictly atomic scaled, such as tunneling, also occur in large scale periodic systems, such as the discretized string, with an identical theoretical explanation. As indicated, it is in the context of numerical computing that the phenomena which have suggested the new theory were first brought to light. We show (using the the string of point masses as an example) that the theory applies also to physical (as opposed to numerical) systems and it will apply as a matter of fact to the other periodic structures which are described by similar mathematics. That no such results have been reported so far in the relevant literature in physics may be attributed in part to the fact that some of the most vivid evidence of quantum-like phenomena may be observed in numerical computing, in experiments with nonuniform grids [6-8]: while nonuniform periodic structures are very common in numerical computing, their occurrence in physical systems is quite rare, and the analysis of wave propagation in classical periodic structures has been devoted almost exclusively to piecewise uniform systems [9, 10]. But this absence of precedents coming from physics may also be attributed to the simple fact that physical systems do not allow for experiments to be created and for the corresponding data to be measured with the same ease and accuracy as in numerical computing. There are of course many examples for this role of computing in the discovery and investigation of new mathematical concepts and physical phenomena: see for instance Zabusky [l 1] for an interesting account of the discovery of solitons, which followed the now famous Fermi-Pasta-Ulam (1955) computer experiments with the discrete space analog of a nonlinear vibrating string.