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arXiv:math-ph/0609036 · Journal of Mathematical Physics

Analytic plane wave solutions for the quaternionic potential step

Stefano De Leo, Gisele Cristina Ducati, Tiago Marques Madureira

Published Aug 1, 2006• 34 citations
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Abstract

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).

DOI
10.1063/1.2227635

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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).
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.
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Abstract. 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 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. PACS. 03.65.Fd – 03.65.Nk I.
Abstract In this chapter we discuss a number of methods for handling stationary state problems in quaternionic quantum mechanics and then consider more generally methods in which the phase of the wave function plays a primary role. We first describe exact methods for reducing the anti Hermitian quaternionic Hamiltonian H to a complex Hermitian Hamiltonian H for an equivalent complex quantum mechanics problem. We then develop stationary state perturbation theory, and as an application compute the leading approximation to the Hamiltonian phase operator IH. We also give a quaternionic analog of the Rayleigh-Ritz variational principle. We proceed next to an extended discussion of the geometric phase in quaternionic quantum mechanics, in both the adiabatic and the nonadiabatic cases. Finally, we give the quaternionic generalization of the eikonal or WKB approximation.
Solving the Schrödinger equation in quaternionic quantum mechanics (QQM) has been a nut to crack. In complex quantum mechanics (CQM), approximation methods have been used to find the solutions of the Schrödinger equation when exact solutions are difficult to obtain. This study challenges the earlier Eikonal (or WKB) approximation formalism [S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields (Oxford University Press, New York, 1995)] developed for an imaginary quaternion potential. In this paper, the correct form of coupled equations, obeying the noncommutative structure and approximated solution for quaternionic time-independent Schrödinger equation (QTISE), has been obtained.