Minimally relativistic Schrödinger equation and the linearly rising potential

Miloslav Znojil

The well known quasi-exact solvability of Schrödinger equation with Coulomb-like potentials is shown to survive a minimally relativistic improvement of the kinetic energy operator .

N. I. Ionescu-Pallas

Gastón García‐Calderón, Alberto Rubio, Jorge Villavicencio

We consider exact time-dependent analytic solutions to the Schr\"odinger equation for tunneling in one dimension with cutoff wave initial conditions at $t=0.$ We obtain that as soon as $t\ensuremath{\ne}0$ the transmitted probability density at any arbitrary distance rises instantaneously with time in a linear manner. Using a simple model we find that the above nonlocal effect of the time-dependent solution is suppressed by consideration of low-energy relativistic effects. Hence at a distance ${x}_{0}$ from the potential the probability density rises after a time ${t}_{0}{=x}_{0}/c$ restoring Einstein causality. This implies that the tunneling time of a particle can never be zero.

Frank V. Kowalski

John R. Fanchi

Alfred Tang, Daniel R. Shillinglaw, George Nill

The non-relativistic Schrodinger equation with the linear and Coulomb potentials is solved numerically in configuration space using the relaxation method. The numerical method presented in this paper is a plain explicit Schrodinger solver which is conceptually simple and is suitable for advanced undergraduate research.