Wigner functions for electron quantum states in a Coulombian potential (hydrogen atom) and two-mode quantum correlations for quasi-Gaussian quantum superpositions

Abstract

This project contemplates the development and use of fundamental structures of Quantum Mechanics described through the Wigner representation in phase space to compute qualitatively and quantitatively the quantum correlation content of two-mode quantum systems (in continuous variables) associated to known solutions of the electron in a Coulomb potential of the Hydrogen atom Coulombian hydrogen atom, not yet widely discussed in phase space. In the first stage of the project, the results for the Wigner functions of known quantum systems, for example, the 1D and 2D harmonic oscillators (the latter, with magnetic coupling), and of the gravitational quantum well described by a linear potential, are completely re-obtained. Once dominated, the analytical strategy shall be applied, in the second step, to the Hydrogen radial wave functions with principal quantum number, n, and with definite angular quantum number characterization, l, as to construct eigenstate Wigner functions an corresponding quasi-Gaussian superpositions of these quantum states. In the final stage, it shall be expected that calculations of quantum (Gaussian) correlations, according to several quantifying measurements, can introduce novel elements for discussing the information content of such quantum systems. In this final stage, preparatory to the student's masters program, quantifiers of some quantum and classic correspondence of quasi-Gaussian states constructed in phase space can also be discussed in the context here proposed. (AU)