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Physical Review A

Nonrelativistic quaternionic quantum mechanics in one dimension

A. J. Davies, Bruce H. J. McKellar

Published Oct 1, 1989• 80 citations
Publisher

Abstract

We present a formalism for treating one-dimensional problems in quaternionic quantum mechanics. As an example, we derive an explicit form for the T matrix for scattering from a square (quaternionic) barrier, and use this result to calculate the transmission and reflection coefficients. We show that the qualitative form of these coefficients is the same as in complex quantum mechanics, even when the barrier has a nonzero value for the quaternionic components of the potential.

DOI
10.1103/physreva.40.4209

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We present a formalism for treating one-dimensional problems in quaternionic quantum mechanics. As an example, we derive an explicit form for the T matrix for scattering from a square (quaternionic) barrier, and use this result to calculate the transmission and reflection coefficients. We show that the qualitative form of these coefficients is the same as in complex quantum mechanics, even when the barrier has a nonzero value for the quaternionic components of the potential.
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.
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.
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).
Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).