Oscillateur harmonique quantique dans un espace de Hilbert bicomplexe et algebre lineaire bicomplexe.

摘要

This thesis presents the work done during my master's degree on bicomplex quantum mechanics. Bicomplex numbers are one of the possible generalizations of complex numbers and they possess the algebraic structure of a commutating ring. The initial goal of this master's degree was to build a structure analogous to the Hilbert spaces in standard quantum mechanics, but standing on bicomplex numbers instead of complex ones. This bicomplex Hilbert space, and the operators acting on it, form what we can call a bicomplex quantum mechanics and we wanted to use it to solve the problems of the quantum harmonic oscillator and hydrogen atom. However, some mathematical difficulties arised on the way and forced us to abandon the hydrogen atom problem. The solution of these difficulties allowed us to write two papers on bicomplex Hilbert spaces and bicomplex linear algebra [1, 2].;This thesis, by articles, explores the algebraic solution to the harmonic oscillator problem in the framework of bicomplex quantum mechanics [3] as well as many results and mathematical tools that we developed during this investigation.