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Multicomplex Wave Functions for Linear And Nonlinear Schrodinger Equations

机译：用于线性和非线性Schrodinger方程的多彗星波函数

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摘要

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.
机译：我们考虑具有一般标量电位的多彗星Schrodinger方程，标准Schrodinger方程和Rochon和Tremblay的Bicomplex Schrodinger方程的概括，用于波浪函数映射到。我们以递归形式确定等效的实值系统，并得出相关的连续性方程，以证明概率保护（标准量子力学的标志）保持在多聚合物中的概率。从这里，我们获得了真实模量，并演示了概率密度为出生的出生的公式。然后，我们注意多种式字路式Schrodinger方程的可能概括，例如用多种式字路上替换标量电位的情况，或者潜在涉及波浪函数的真实模量的情况，导致多种式字非线性Schrodinger方程。最后，为了展示这种等式的解决方案方法，我们获得了多种式用途的特定解决方案。我们在量子力学的标准结果的上下文中解释了广义的结果。

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