Multicomplex Wave Functions for Linear And Nonlinear Schrödinger Equations

Abstract

We consider a multicomplex Schrödinger equation with general scalar potential, a generalization of both the standard Schrödinger equation and the bicomplex Schrödinger equation of Rochon and Tremblay, for wave functions mapping onto C k. We determine the equivalent real-valued system in recursive form, and derive the relevant continuity equations in order to demonstrate that conservation of probability (a hallmark of standard quantum mechanics) holds in the multicomplex generalization. From here, we obtain the real modulus and demonstrate the generalized multicomplex version of Born's formula for the probability densities. We then turn our attention to possible generalizations of the multicomplex Schrödinger equation, such as the case where the scalar potential is replaced with a multicomplex-valued potential, or the case where the potential involves the real modulus of the wave function, resulting in a multicomplex nonlinear Schrödinger equation. Finally, in order to demonstrate the solution methods for such equations, we obtain several particular solutions to the multicomplex Schrödinger equation. We interpret the generalized results in the context of the standard results from quantum mechanics.

Algorithm 1008

“…The method will be applied to obtain sensitivities when accurate derivatives of a scalar valued functions f : R n → R, with respect to many inputs are required. There are other potential areas of application of multicomplex numbers which will be facilitated by the development of the class described here such as in quantum mechanics [31].…”

Multicomplex solitons

“…Hyperbolic extensions of the complex Hilbert space have been studied in [9]. The standard Schrödinger equation was bicomplexified in [10] and further studied in [11,12,13,14]. Quaternionic and coquaternionic quantum mechanics and quantum field theory have been studied for a long time, see e.g.…”