A hydrogen-like atom confined within an impenetrable spherical box

Abstract

Ground- and excited-state energies and wavefunctions of a hydrogen-like atom, confined at the centre of a spherical `box' with impenetrable walls, are derived using a variety of analytical and algebraic methods. In particular, asymptotic forms (which yield highly accurate energies) are obtained for the case of large box radii, and departures from the Coulomb degeneracy for a box of finite radius demonstrated. For smaller boxes, economical wavefunctions are developed on the basis of unconventional forms of Rayleigh-Schrödinger perturbation theory, and of a Lie algebraic treatment of a transformed Schrödinger equation.

Dynamic dipole polarizabilities of the ground and excited states of confined hydrogen atom computed by means of a mapped Fourier grid method

“…The latter is highly accurate. The accuracy of computed energy eigenvalues is of the order of 10 decimal places or better, as judged by their comparison with those reported by other authors [5,6,9]. Knowing the level energies, we may calculate the relevant resonant frequencies.…”

“…All other ns state contributions are absorptive, including the one due to 2s, since for confined hydrogen, this level lies higher in energy than 2p. Moreover, the same 2p3nd transitions appearing in ␣ 2p͉m͉ϭ1 () are also evident in ␣ 2pmϭ0 () with an exception for R ϭ 2 a.u., where they are superimposed with the 2p3ns, n Ն 2 resonances because the ns and (n ϩ 1)d, n Ն 2 levels are degenerate in this case [5,6]. The static polarizability ␣ 2pmϭ0 ( ϭ 0) is always positive and decreasing with decreasing R but for large confinement radii diverges, acquiring much higher values 1 Dominating absorptive terms are recognized by the fact that the dynamic polarizabilities are positive on the red side and negative on the blue side of the resonant singularities.…”

“…As the radius of confinement decreases, the higher angular momentum ᐉ-states get relatively more stabilized and the accidental degeneracy breaks down in such a way that E 2p Ͻ E 2s , E 3d Ͻ E 3p Ͻ E 3s , etc. Groundand excited-state energies and wavefunctions of a hydrogen-like atom, confined at the center of a spherical "box" with impenetrable walls, have been derived using a variety of analytical and algebraic methods [5][6][7][8][9]. Static dipole polarizabilities have also been reported for the ground and some excited states of confined hydrogen [7][8][9][10][11].…”

Energy levels and structural properties of compressed hydrogen atom under Debye screening

“…Sommerfeld & Welker (1938) carried out investigations on the energy levels of hydrogen in a spherical box. Subsequently, a number of calculations (Ley-Koo 1979;Ludena 1977Ludena , 1978Marin & Cruz 1991a, 1991bZicovich-Wilson 1994;Fowler 1984;Aquino 1995;Dineykhan 1999;Singh 1984;Montgomery 2002;Laughlin 2002) have been performed for studying the energy levels of compressed atoms. In the current communication we have performed a systematic analysis of the effect of a plasma screening and that of a finite confining radius on the dipole polarizabilities and 2p, 3p and 4p energy levels for the hydrogen atom, their oscillator strengths and transition probabilities with a view to estimate the order of magnitude of the spectral line shifts under such confinements.…”

“…Very recently the dynamic dipole polarizability of compressed hydrogen was calculated by Montgomery (2002) for different radii of compression and 1s→2p transition energy was determined. The recent calculation of Laughlin et al (2002) on compressed hydrogen atom using a variety of analytical and algebraic methods yields accurate estimate of the ground and excited state energies and wave functions. A brief description of the current method is described in Sect.…”