Exact solutions for spherically confined hydrogen‐like atoms

Abstract

We present exact analytical solutions for the much-studied problem of a hydrogen-like atom confined in a spherical box of radius R. These solutions, which are obtained for all states and all R, are expressed directly in terms of the Kummer M-functions whose analytical and numerical properties are well known, and may be calculated using standard computing packages. The solutions are illustrated by precise calculations that yield accurate energies E for any given radius R, or for R when E is known. In the special case where E ϭ 0, it is shown that the solution may be expressed in terms of Bessel functions. Finally, the physical assumptions made in applying this model to describe atomic confinement are discussed critically.

Studies on the 3D confined potentials using generalized pseudospectral approach

“…The two sets of results are found to be in excellent agreement .In Table 2 , we have similarly compared the results of our calculations of the energy levels of the SCHA with the most accurate numerical data reported earlier 10 . Once again , the two sets of results agree almost exactly with each other .It is concluded from Table 1-2 that the GPS method is able to describe the energy levels of the spherically confined systems with remarkable accuracy, as found earlier also for the unconfined systems [35][36][37][38] .The presently calculated energies are significantly more accurate than all the other hitherto reported results for a large number of n , ℓ , R c values .Consequently, the scope of application of the GPS method is hereby extended to cover a large variety of the problems in chemical physics enumerated by Chu 35 from free to the confined systems with equal ease and accuracy .…”

“…In particular, it has been observed that the degeneracy and relative ordering of energy levels of the unconfined system are both influenced significantly under the effect of confining potentials. Considerable theoretical efforts have recently been made in performing more accurate computations on simple model systems like the hydrogen 10 and helium atom 11 which could also serve as a benchmark for the approximate methods. For all angular momentum values, the exactly solvable problems of the free (unconfined) hydrogen atom (FHA) and the free isotropic harmonic oscillator (FIHO), respectively, define the standard text book problems in quantum mechanics .Under the conditions of spherical confinement, on the other hand, while the spherically confined hydrogen atom (SCHA) has been studied in considerable detail, the spherically confined isotropic harmonic oscillator (SCIHO) remains to be studied similarly.…”

Exact solutions for shell-confined hydrogen-like atoms: polarisabilities and Shannon entropies

“…In Table 3 we again use R 1 ¼ 2, Z ¼ 1 but for the first excited l ¼ 0 state and for the larger values of R 2 the energies are small and negative, but eventually they become positive and increase sharply. This is similar to the behaviour in our earlier work where the confinement was purely spherical (R 1 ¼ 0 for some R 2 ) [1,3]. To deal with the case of positive energy we may replace by i ( real) and solve (13), equating both real and imaginary parts to zero simultaneously.…”

“…The function f(c, d, 2r) satisfies the same second-order differential equation studied previously, [1] , [2] but with two-point finite boundary conditions…”

“…The model consists of the conventional Coulomb potential restricted to a finite radial interval R 1 5r5R 2 . In previous work [1][2][3] we adopted the traditional model with R 1 ¼ 0; the present extension introduces nontrivial changes in the mathematical treatment of an exactly soluble problem.…”