9 research outputs found
High Resolution Sharp Computational Methods for Elliptic and Parabolic Problems in Complex Geometries
Transition probabilities of forbidden lines within the electronic configurations 6p 2 and 6p 7p of Pb I
Falling of a quantum particle in an inverse square attractive potential
Evolution of a quantum particle in an inverse square potential is studied by analysis of
the equation of motion for 〈r2〉. In such a way we identify the conditions of falling of
a particle into the center. We demonstrate the existence of a purely quantum limit of
falling, namely, a particle does not fall, when the coupling constant is smaller than a
certain critical value. Also the time of falling of a particle into the center is
estimated. Although there are no stationary energy levels for this potential, we show that
there are quasi-stationary states which evolve with 〈r2〉 being constant in time. Our results are compared with
measurements of neutral atoms falling in the electric field of a charged wire.
Modifications of the experiment, which may help in observing quantum limit of falling, are
proposed
Bound States of Energy Dependent Singular Potentials
We consider attractive power-law potentials depending on energy through their coupling constant. These potentials are proportional to 1/|x| m with m ≥ 1 in the D = 1 dimensional space, to 1/r m with m ≥ 2 in the D = 3 dimensional space. We study the ground state of such potentials. First, we show that all singular attractive potentials with an energy dependent coupling constant are bounded from below, contrarily to the usual case. In D = 1, a bound state of finite energy is found with a kind of universality for the eigenvalue and the eigenfunction, which become independent on m for m > 1. We prove the solution to be unique. A similar situation arises for D = 3 for m > 2, except that, in this case, the solution is not directly comparable to a bound state: the wave function, though square integrable, diverges at the origin