Studies on some singular potentials in quantum mechanics

Abstract

A simple methodology is suggested for the efficient calculation of certain central potentials having singularities. The generalized pseudospectral method used in this work facilitates {\em nonuniform} and optimal spatial discretization. Applications have been made to calculate the energies, densities and expectation values for two singular potentials of physical interest, {\em viz.,} (i) the harmonic potential plus inverse quartic and sextic perturbation and (ii) the Coulomb potential with a linear and quadratic term for a broad range of parameters. The first 10 states belonging to a maximum of $\ell=8$ and 5 for (i) and (ii) have been computed with good accuracy and compared with the most accurate available literature data. The calculated results are in excellent agreement, especially in the light of the difficulties encountered in these potentials. Some new states are reported here for the first time. This offers a general and efficient scheme for calculating these and other similar potentials of physical and mathematical interest in quantum mechanics accurately

Ro-vibrational studies of diatomic molecules in a shifted Deng-Fan oscillator potential

“…In this section, we briefly outline the GPS formalism for solving the nonrelativistic Schrödinger equation of a Hamiltonian containing a DF potential. Only the essential steps are given; relevant details may be found in previous works and the references therein). Atomic units are used throughout the article, unless otherwise mentioned.…”

“…Although, some decent results are available, there is a need of systematic analysis in terms of accurate eigenvalues, eigenvalues and other relevant quantities, especially for the situations mentioned earlier. For this, we use the generalized pseudospectral method (GPS), which has been very successful for a number of physically important quantum systems, such as, spiked harmonic oscillator, rational, Hulthén, Yukawa, logarithmic, power‐law, Morse potential as well as static and dynamic studies in molecules and atoms (including Rydberg states) and so forth . Thus, at first, a detailed study is presented on the bound states of DF potential covering 36 states corresponding to vibrational quantum number up to n = 7.…”

Point-particle effective field theory I: classical renormalization and the inverse-square potential

“…In that case, we define 1 as ∞, since the lack of higher angular momentum state means there is no upper bound on α. 8 Unless the physics of the source does allow it to be a sink -or source -of probability, such as in situations like those described in [13], for example. 9 Strictly speaking the reasoning presented here is only true for delta-function interactions in the absence of inversesquare potentials, because the singularity of the −g/r 2 potential at r = 0 undermines the argument that only the derivatives and delta function can contribute when integrating the equations of motion over a small pillbox.…”

Studies on the Bound-State Spectrum of Hyperbolic Potential

“…Since this has been already discussed earlier [19][20][21][22][23][24][25][26], it suffices here to give only a review.…”

“…However for the potential under consideration, relatively much less effort has been made. In this study, we employ the generalized pseudospectral (GPS) method, found to offer promising results for a number of physical situations (see, for example, [19][20][21][22][23][24][25][26] and references therein) in recent years. This is a simple, efficient and yet accurate method which relies on an optimal, spatial discretization Then a detailed analysis of energy variation with respect to the parameters α, σ 0 , D e is presented, which, to our knowledge, has not been attempted before.…”