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Advances in Applied Clifford Algebras

, Volume 14, Issue 2, pp 231–248 | Cite as

Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

  • D. Rochon
  • S. Tremblay
Original Paper

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.

Keywords

Wave Function Quantum Mechanics Continuity Equation Commutative Ring Discrete Symmetry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  1. 1.Département de mathématiques et d’informatiqueUniversité du Québec à Trois-RivièresTrois-RivièresCanada

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