Exact quasi-relativistic wavefunctions of Hydrogen-like atoms

Abstract

Exact solutions of a novel quasi-relativistic quantum mechanical wave equation are found for Hydrogen-like atoms. This includes both, an exact analytical expression for the energies of the bound states, and exact analytical expressions for the wavefunctions, which successfully describe quantum particles with mass and spin-0 up to energies comparable to the energy associated to the mass of the particle. These quasi-relativistic atomic orbitals may be used for improving ab-initio software packages dedicated to numerical simulations in physical-chemistry and atomic and solid-state physics.

Direct Relativistic Extension of the Madelung-de-Broglie-Bohm Reformulations of Quantum Mechanics and Quantum Hydrodynamics

“…The three-dimensional (3D) GdeP equation for a particle moving at relativistic speeds in a potential V is given by the following expression [13][14][15]:…”

“…(1) is well-defined, but it is advantageous to write the operator K as in Eq. (1), because this results in an equation formally like the Schrödinger equation, which can then be exactly solved following similar procedures than the ones required for solving the Schrödinger equation [11][12][13][14][15]. It is also the striking similarity between Eq.…”

“…(2). This may look at first as an unmanageable problem; however, this is not the case in at least several interesting cases [11][12][13][14][15]. In general, Eq.…”

“…(1) and its solutions, and detailed discussions of how to solve Eq. (1) for some interesting potentials V, can be found in recently published works [11][12][13][14][15]. In a nutshell, solving Eq.…”

“…Other general approaches have been reported [10], but we explore in this work an alternative methodology for extending, to the relativistic domain, the known non-relativistic quantum hydrodynamics and quantum theories with trajectories. Our approach is based in a surprising wave equation which resembles the Schrödinger equation, but describes a particle with mass and spin-0 which moves through a potential V, and has the correct relativistic relation between the linear momentum p and the kinetic energy K [11][12][13][14][15]:…”