Multicomplex wave functions for linear and nonlinear Schrödinger equations

Journal article

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 (Formula presented.). 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.

Authors: Theaker, K; van Gorder, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Advances in Applied Clifford Algebras
• Volume: 27
• Issue: 2
• Pages: 1857–1879
• Publication date: 2016-11-01
• Acceptance date: 2016-10-22
• DOI: 10.1007/s00006-016-0734-2
• EISSN: 1661-4909
• ISSN: 0188-7009
• Keywords: Multicomplex analysis; Schrödinger equation; Multicomplex number system