Spherical confinement of coulombic systems inside an impenetrable box: H atom and the Hulthén potential

Abstract

The generalized pseudospectral method is employed to study spherical confinement in two simple Coulombic systems: (i) well celebrated and heavily studied H atom (ii) relatively less explored Hulthén potential. In both instances, arbitrary cavity size, as well as low and higher states are considered. Apart from bound state eigenvalues, eigenfunctions, expectation values, quite accurate estimates of the critical cage radius for H atom for all the 55 states corresponding to n ≤ 10, are also examined. Some of the latter are better than previously reported values. Degeneracy and energy ordering under the isotropic confinement situation are discussed as well. The method produces consistently high-quality results for both potentials for small as well as large cavity size.For the H atom, present results are comparable to best theoretical values, while for the latter, this work gives considerably better estimates than all existing work so far.

Critical parameters and spherical confinement of H atom in screened Coulomb potential

“…Ten selected r c , viz ., 0.1, 0.5, 1, 2, 5, 10, 20, 30, 50, and 100 a.u., are chosen to scan the whole range of confinement. Similar to the confined H atom or Hulthén potential case, for a given δ c , starting from a large positive value at smaller r c , energy steadily decreases monotonically, crossing zero and becoming negative at certain r c , eventually approaching a constant value at a sufficiently large r c thereafter. Available reference energies are provided at four r c , viz., 2, 5, 10, and 50, respectively, which seem to match rather well with present energies.…”

“…So for a given n , the orderings are found as: ; ; , etc. Similar energy orderings were found in confined H atom and Hulthén potential . As enclosure size is reduced, numerous complex energy splitting is observed, especially for states having high n , ℓ quantum numbers.…”

“…The GPS method has been shown to be a simple useful and powerful approach for a variety of physical systems. They were discussed previously in a number of communications; hence not repeated here. Spherical confinement of a particle in a central potential is modeled, without any loss of generality, by the following radial non‐relativistic Schrödinger equation (atomic unit employed unless otherwise mentioned): where n , ℓ signify usual radial and angular quantum numbers, while v ( r ) characterizes the particular screening potential under consideration.…”

Information‐entropic measures in free and confined hydrogen atom

“…Effect of compression on energy levels of ground and excited states, as well as other properties like hyperfine splitting constant, dipole shielding factor, nuclear magnetic screening constant, pressure, static, and dynamic polarizability, were examined . Numerous theoretical methods varying in complexity, sophistication were employed; a selected set includes perturbation theory, Padé approximation, WKB method, Hypervirial theorem, power‐series solution, super‐symmetric quantum mechanics, Lie algebra, Lagrange‐mesh method, asymptotic iteration method, generalized pseudo‐spectral method, and so forth and references therein. Exact solutions are expressible in terms of Kummer M‐function (confluent hypergeometric).…”

“…Two prototypical systems receiving maximum attention are harmonic oscillator (in 1D, 2D, 3D, D dimension) [9][10][11][12][13][14] and confined hydrogen atom (CHA) inside a spherical enclosure. [3,10,[15][16][17][18][19][20][21][22][23][24] A CHA within an impenetrable (as well as penetrable) cavity was studied quite vigorously leading to a host of attractive properties-both from physical and mathematical point of view. They offer many unique phenomena, especially relating to simultaneous, incidental and interdimensional degeneracy.…”