Bicomplex quantum mechanics: I. The generalized Schrödinger equation

Details Bicomplex quantum mechanics: I. The generalized Schrödinger equation Bicomplex quantum mechanics: I. The generalized Schrödinger equation

2007-09-20

arxiv.org/abs/0709.3242v1

Adv. Appl. Cliff. Alg., 14, No 2 (2004) 231-248

Physics

Quantum Physics

Scientific paper

We introduce the set of bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0+w_1 \bold{i_1}+w_2\bold{i_2}+w_3 \bold{j}| w_0,w_1,w_2,w_3 \in \mathbb{R}\}$ where $\bold{i^{\text 2}_1}=-1, \bold{i^{\text 2}_2}=-1, \bold{j}^2=1,\ \bold{i_1}\bold{i_2}=\bold{j}=\bold{i_2}\bold{i_1}$. We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr\"odinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr\"odinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symetries. We obtain the standard Born's formula for the class of bicomplex wave functions having a null hyperbolic angle.

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