Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

Bicomplex quantum mechanics: I. The generalized Schrödinger equation

We introduce the set of bicomplex numbers T which is a commutative ring with zero divisors defined by T = {w0 + w1i1 + w2i2 + w3j| w0, w1, w2, w3 ∈ R} where i21 = −1, i22 = −1, j = 1, i1i2 = j = i2i1. 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 ...

Bicomplex Quantum Mechanics: II. The Hilbert Space

Using the bicomplex numbers T which is a commutative ring with zero divisors defined by T = {w0+w1i1+w2i2+ w3j | w0, w1, w2, w3 ∈ R} where i21 = −1, i22 = −1, j = 1, i1i2 = j = i2i1, we construct hyperbolic and bicomplex Hilbert spaces. Linear functionals and dual spaces are considered on these spaces and properties of linear operators are obtain; in particular it is established that the eigenv...

Quantum Mechanics as Asymptotics of Solutions of Generalized Kramers Equation

We consider the process of diffusion scattering of a wave function given on the phase space. In this process the heat diffusion is considered only along momenta. We write down the modified Kramers equation describing this situation. In this model, the usual quantum description arises as asymptotics of this process for large values of resistance of the medium per unit of mass of particle. It is ...

Solving Schrödinger equation via Tartaglia/Pascal triangle: a possible link between stochastic processing and quantum mechanics

In a recent paper [1] it was shown a clean connection between the solution of the classical heat equation and the Tartaglia/Pascal/Yang-Hui triangle. When the time variable in the heat equation is substituted with the imaginary time, the heat equation becomes the Schrödinger equation of the quantum mechanics. So, a conjecture was put forward about a connection between the solution of the Schröd...

Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation.