Spherical confinement of coulombic systems inside an impenetrable box: H atom and the Hulthén potential (original) (raw)
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Critical parameters and spherical confinement of H atom in screened Coulomb potential
International Journal of Quantum Chemistry, 2016
Critical parameters in three screened potentials, namely, Hulthén, Yukawa and exponential cosine screened Coulomb potential are reported. Accurate estimates of these parameters are given for each of these potentials, for all states having n ≤ 10. Comparison with literature results is made, wherever possible. Present values compare excellently with reference values; for higher n, ℓ, our results are slightly better. Some of these are presented for first time. Further, we investigate the spherical confinement of H atom embedded in a dense plasma modeled by an exponential cosine screened potential. Accurate energies along with their variation with respect to box size and screening parameter are calculated and compared with reference results in literature. Sample dipole polarizabilities are also provided in this case. The generalized pseudospectral method is used for accurate determination of eigenvalues and eigenfunctions for all calculations.
2020
The orbital, ground and excited state energies of many electron atoms confined by an impenetrable spherical cavity with radius R are calculated using Quantum Genetic Algorithm (QGA) approach and Hartree-Fock Roothaan (HFR) theory. The important properties such as static and dynamic polarizability, oscillator strength and static pressure are investigated as perturbative. The results reveal that cavity radius and impurity charge have played an important role on the polarizability, the oscillator strength and pressure of the system. In addition, it is seen that when cavity radius is extremely large, all energies and the other physical parameters approach the energies and physical parameters of unconfined atom. As the dot radius decreases, the polarizability of system because of the strong spatial confinement decreases, but the pressure exerting on the system as the cavity radius R is shrunk increases. In addition, as the impurity charge increases, the magnitude of the oscillator strength decreases.
A study of the confined hydrogen atom using the finite element method
Journal of Physics B: Atomic, Molecular and Optical Physics, 2005
The hydrogen atom confined by an infinite spherical potential barrier is studied employing a variational procedure based on the p-version of the finite element method. In such a procedure, the spherical spatial confinement is imposed straightforwardly by removing a local basis function. The calculations have been performed for estimating the energy spectrum, the dipole polarizability and the effective pressure for various confinement radii. The effect of the spatial confinement on these quantities is analysed. The results obtained are compared with those previously published in the literature and the efficiency of the finite element method to treat confined quantum systems is discussed.
Three-Dimensional Confinement: WKB Revisited
Journal of Mathematical Chemistry, 2000
An alternate formalism is developed to determine the energy eigenvalues of quantum mechanical systems, confined within a rigid impenetrable spherical box of radius r 0 , in the framework of Wentzel-Kramers-Brillouin (WKB) approximation. Instead of considering the Langer correction for the centrifugal term, the approach adopted here is that of Hainz & Grabert : The centrifugal term is expanded perturbatively (in powers ofh), decomposing it into 2 terms-the classical centrifugal potential and a quantum correction. Hainz and Grabert found that this method reproduced the exact energies of the hydrogen atom, to the first order inh, with all higher order corrections vanishing. In the present study, this formalism is extended to the case of radial potentials under hard wall confinement, to check whether the same argument holds good for such confined systems as well. As expicit examples, 3 widely known potentials are studied, which are of considerable importance in the theoretical treatment of various atomic phenomena involving atomic transitions, viz., the 3-dimensional Harmonic oscillator, the hydrogen atom and the Hulthen potential.
Studies on the 3D confined potentials using generalized pseudospectral approach
Physics Letters A, 2006
In presence of the spherically confined three-dimensional potentials with impenetrable boundaries, the generalized pseudospectral method is shown to provide accurate eigenvalues, eigenfunctions, and radial expectation values for (a) the isotropic harmonic oscillator, (b) the H atom and (c) the Davidson oscillator. Several novel degeneracy conditions are obtained for (a) when the radius of confinement is suitably chosen at the radial nodes corresponding to the free states.
HYDROGEN ATOM IN SPACE WITH A COMPACTIFIED EXTRA DIMENSION AND POTENTIAL DEFINED BY GAUSS' LAW
We investigate the consequences of one extra spatial dimension for the stability and energy spectrum of the non-relativistic hydrogen atom with a potential defined by Gauss' law, i.e. proportional to 1/|x| 2. The additional spatial dimension is considered to be either infinite or curled-up in a circle of radius R. In both cases, the energy spectrum is bounded from below for charges smaller than the same critical value and unbounded from below otherwise. As a consequence of compactification, negative energy eigenstates appear: if R is smaller than a quarter of the Bohr radius, the corresponding Hamiltonian possesses an infinite number of bound states with minimal energy extending at least to the ground state of the hydrogen atom.
2019
Considering the stationary Schrödinger equation for a general pseudo-Coulomb potential as the normal form of the associated Laguerre equation, we review, in one and three dimensions, the bound-state solutions for the potential, when the inverse-square-term coupling is not less than a negative critical value. We show that, as a consequence of the inverse-square-term coupling being a two-to-one mapping for all but one of the allowed negative values of its parameter, an additional sequence of bound-state energies emerges for each of the respective potentials. In this framework, the slightest relaxation of the boundary condition for the radial wave function at the origin results in minus-infinity ground-state energy for the Coulomb potential, rendering the hydrogen atom unstable. The article has been published in the journal "New Horizons in Mathematical Physics", Vol. 3, No. 4, December 2019. Available at: https://dx.doi.org/10.22606/nhmp.2019.34001
Bound States in Coulomb Systems‐Old Problems and New Solutions
2012
We analyze the quantum statistical treatment of bound states in Hydrogen considered as a system of electrons and protons. Within this physical picture we calculate isotherms of pressure for Hydrogen in a broad density region and compare to some results from the chemical picture. First we resume in detail the two transitions along isotherms : (i) the formation of bound states occurring by increasing the density from low to moderate values, (ii) the destruction of bound states in the high density region, modelled here by Pauli-Fock effects. Avoiding chemical models we will show, why bound states according to a discrete part of the spectra occur only in a valley in the T-p plane. First we study virial expansions in the canonical ensemble and then in the grand canonical ensemble. We show that in fugacity representations the population of bound states saturates at higher density and that a combination of both representations provides quickly converging equations of state. In the case of degenerate systems we calculated first the density-dependent energy levels, and find the pressure in Hartree-Fock-Wigner approximation showing the prominent role of Pauli blocking and Fock effects in the selfenergy.
Electron structure of endohedrally confined atoms: atomic hydrogen in an attractive shell
Journal of Physics B: Atomic, Molecular and Optical Physics, 1999
The properties of hydrogen confined endohedrally at the geometrical centre of a spherical, attractive short-range potential shell are explored. The evolution of the energy spectrum, as a function of the depth of the shell, is found to exhibit unusual level crossings and degeneracies resulting in avoided crossings and a new phenomenon of 'mirror collapse' where the localized states switch places. In addition, a new level ordering, principally by the number of nodes in the radial wavefunction, develops. The results apply generally to endohedrally confined atoms. Further, they suggest a new tool for controlling the properties of atoms.