20 research outputs found
Bicomplex Quantum Mechanics: II. The Hilbert Space
Using the bicomplex numbers which is a commutative ring with
zero divisors defined by where , we construct hyperbolic and bicomplex Hilbert spaces.
Linear functionals and dual spaces are considered and properties of linear
operators are obtained; in particular it is established that the eigenvalues of
a bicomplex self-adjoint operator are in the set of hyperbolic numbers.Comment: 25 pages, no figur
Bicomplex quantum mechanics: I. The generalized Schr\"odinger equation
We introduce the set of bicomplex numbers which is a commutative
ring with zero divisors defined by 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
Relationship between the Mandelbrot Algorithm and the Platonic Solids
This paper focuses on the dynamics of the eight tridimensional principal
slices of the tricomplex Mandelbrot set for the power 2: the Tetrabrot, the
Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot,
the Airbrot (octahedron) and the Firebrot (tetrahedron). In particular, we
establish a geometrical classification of these 3D slices using the properties
of some specific sets that correspond to projections of the bicomplex
Mandelbrot set on various two-dimensional vector subspaces, and we prove that
the Firebrot is a regular tetrahedron. Finally, we construct the so-called
"Stella octangula" as a tricomplex dynamical system composed of the union of
the Firebrot and its dual, and after defining the idempotent 3D slices of
, we show that one of them corresponds to a third Platonic
solid: the cube
Sur une généralisation des nombres complexes : les tétranombres
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal