Quantum mechanics on phase space and the Coulomb potential

Abdelmadjid Maireche

In this recently work, we have obtained the analytical solutions of the modified Schrödinger equation (MSE) with new Coulomb potentials plus linear and harmonic radial terms (NCPLHRT) for hydrogenic atoms in noncommutative 3-dimensional real space-phase (NC: 3D-RSP). We applied, the generalized Bopp’s shift method and standard perturbation theory in the framework of two infinitesimal parameters due to (space-phase) noncommutativity, we obtained new energy eigenvalues, which depended with discrete atomic quantum numbers and parameters of studied potential, in addition to the corresponding new Hamiltonian operator. Our research also conveys another innovative feature, the global group symmetry (NC: 3D-RSP) were broken simultaneously and replaced by the residual local subgroup (NC: 3D-RS) under interaction of hydrogenic atoms with NCPLHRT.

Manuel Gadella

Waldemar Wolney Filho

In this work we are presenting a modified Coulomb potential function to describe the interaction between two micro-scopic electric charges. In particular, concerning the interaction between the proton and the electron in the hydrogen atom. The modified potential function is the product of the classical Coulomb potential and an oscillatory function dependent on a quantized phase factor. The oscillatory function picks up only selected points along the Coulomb potential, creating potential wells and barriers around the nucleus of the atom. The new potential reveals us new features of the hydrogen atom. Searching for a manner to determine the phase factor, we are using the concept of the de Broglie particle wavelike behavior and the quantum analogue of the virial theorem for describing the bound motion of a particle in a central force field. This procedure is a kind of feedback action, where we are making use of well established concepts of the quantum mechanics aiming to determine the phase factor of the new interaction potential.

Franklin E. Schroeck

U. Oseguera, MANUEL DE LLANO

The quantum mechanical problem of a particle moving in one dimension (1D) under the influence of a singular potential is reexamined. An attractive delta function potential of infinite strength and the 1D Coulomb potential is considered in particular. In both cases the singularity acts as an impenetrable barrier, thus dividing space into two independent regions. Analysis of the scattering states gives results in agreement with this conclusion. The implications that the bound states of the 1D Coulomb potential cannot be classified according to parity, even though the potential is symmetric, is also discussed herein.

Nicolaas P. Landsman