Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

Abstract.

We introduce the set of bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T} = \{ \omega _0 + \omega _1 {\mathbf{i}}_{\mathbf{1}} + \omega _2 {\mathbf{i}}_{\mathbf{2}} + \omega _3 {\mathbf{j}}|\omega _0 ,\omega _1 ,\omega _2 ,\omega _3 \in \mathbb{R}\}$ where ${\mathbf{i}}_{\mathbf{1}}^2 = - 1,\;{\mathbf{i}}_{\mathbf{2}}^2 = - 1,\; {\mathbf{j}}^2 = 1,\;{\mathbf{i}}_{\mathbf{1}} {\mathbf{i}}_{\mathbf{2}} = {\mathbf{j}} = {\mathbf{i}}_{\mathbf{2}} {\mathbf{i}}_{\mathbf{1}} .$ We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schrödinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schrödinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symmetries. We obtain the standard Born’s formula for the class of bicomplex wave functions having a null hyperbolic angle.