Multicomplex Wave Functions for Linear And Nonlinear Schrodinger Equations

We consider a multicomplex Schrodinger equation with general scalar potential, a generalization of both the standard Schrodinger equation and the bicomplex Schrodinger equation of Rochon and Tremblay, for wave functions mapping onto . 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 Schrodinger 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 Schrodinger equation. Finally, in order to demonstrate the solution methods for such equations, we obtain several particular solutions to the multicomplex Schrodinger equation. We interpret the generalized results in the context of the standard results from quantum mechanics.