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arXiv:1012.4613 · Journal of Mathematical Physics

A closed formula for the barrier transmission coefficient in quaternionic quantum mechanics

Stefano De Leo, Gisele Cristina Ducati, Vinícius Leonardi, Kenia Pereira

Published Nov 1, 2010• 15 citations
arXivHTMLPDF

Abstract

In this paper, we analyze, by using a matrix approach, the dynamics of a nonrelativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrödinger equation allows us to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of nonrelativistic particle was motivated by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics and by using the stationary phase method, gives the possibility to discuss transmission times.

DOI
10.1063/1.3504165

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In this paper, we analyze, by using a matrix approach, the dynamics of a nonrelativistic particle in presence of a quaternionic potential barrier. The matrix method used to solve the quaternionic Schrödinger equation allows us to obtain a closed formula for the transmission coefficient. Up to now, in quaternionic quantum mechanics, almost every discussion on the dynamics of nonrelativistic particle was motivated by or evolved from numerical studies. A closed formula for the transmission coefficient stimulates an analysis of qualitative differences between complex and quaternionic quantum mechanics and by using the stationary phase method, gives the possibility to discuss transmission times.
By using the recent mathematical tools developed in quaternionic differential operator theory, we solve the Schrödinger equation in the presence of a quaternionic step potential. The analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart. A brief discussion on reflected and transmitted times, performed by using the stationary phase method, and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented. The analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem (up to now solved by numerical method).
Beginning with the quaternionic generalization of the Schr\"odinger equation, we present a simple system consisting of two successive square barriers of general (quaternionic) height, and discuss situations in which the predictions of quaternionic quantum mechanics differ from those of the usual complex theory. Specifically, we show that the transmission coefficient may pick up a phase change on reversal of the barrier order. We comment on why this phase change would not necessarily be observed in experiments. Finally, we present a necessary condition for the magnitude of the transmission coefficient to change under reversal of the direction of traversal of the barrier.
At the turn of the twentieth century, the establishment of quantum theory propelled rapid advancements, particularly in the understanding of quantum tunnelling—a fundamental phenomenon in quantum mechanics crucial for various physical processes. The quantum phenomenon of particle passes through potential barriers is of great importance. In classical physics, when the energy of a particle is less than the height of a double barrier structure, it is impossible for it to pass through. However, quantum mechanics allows a particle to penetrate the barrier and emerge on the other side. This paper explores the quantum tunnelling effect, focusing on the single potential barrier model in one dimension and subsequently extending to the double potential barrier model. The Schrödinger equation provides the foundational framework for elucidating the motion of microscopic particles, emphasizing wave-particle duality inherent in quantum mechanics. The analysis of the single potential barrier model involves solving the Schrödinger equation in different regions, determining wave functions and coefficients through boundary conditions. The transmission coefficient is derived, representing the probability of a particle passing through a barrier. In the case of a “thick” barrier, an approximate form for transmission coefficient is provided, demonstrating the exponential decrease in transmission probability with increasing barrier thickness.
In looking for qualitative differences between quaternionic and complex formulations of quantum physical theories, we provide a detailed discussion of the behavior of a wave packet in the presence of a quaternionic time-independent potential step. In this paper, we restrict our attention to diffusion phenomena. For the group velocity of the wave packet moving in the potential region and for the reflection and transmission times, the study shows a striking difference between the complex and quaternionic formulations which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation.