The bound-state energy spectrum and its evolution for a hydrogen atom located along the axis of a standard cylindrical confining cavity with either impenetrable or penetrable confining boundaries are obtained by solving the stationary Schrödinger equation using a finite differences approach. New results are first presented for a nuclear-centered position for the penetrable case as the barrier height and cavity size change. Special attention is then given to the energy-level dependence on the nuclear position along the cylinder axis, where proper identification for the evolution of states from the nuclear-centered position (centered states) up to the cylinder cap (cap states) is proposed, while the corresponding state evolution for intermediate nuclear positions (intermediate states) remains consistent with node conservation and symmetry. It is found that in general the energy levels evolve with an increasing value as the nuclear position is shifted from the central position up to a cylinder cap. As the barrier height (and cavity size) are reduced, a progressive extinction of bound states appears in the order cap states, intermediate states and centered states. Finally, a predominance of cavity-size over barrier-height effects on the energy level shift is found.

Spheroconal harmonics are the natural basis for the description of asymmetric-molecule rotations (Kramers and Ittmann, Zeitschrift für Physik, 1929, 53, 553; Piña, J Mol Str 1999, 493, 159) and also an alternative to the familiar spherical harmonics as the angular part of the Schrödringer equation eigenfunctions for central potentials (Kalnins et al. SIAM J Appl Math 1976, 30, 360). We have dealt with their properties and matrix evaluation in connection with the rotations of asymmetric molecules (Ley-Koo and Méndez-Fragoso, Rev Mex Fís 2008, 54, 162) and the construction of a generating function for the complete wave functions of the Hydrogen atom (Ley-Koo and Góngora, Int J Quantum Chem 2009, 109, 790). For these cases, the spheroconal harmonics are products of Lamé polynomials A n 1 (χ 1 ) B n 2 (χ 2 ) in the respective angular coordinates χ 1 , χ 2 , with n 1 + n 2 = , the angular momentum label. More recently during the investigations of the Hydrogen atom (Méndez-Fragoso and Ley-Koo, Int J Quantum Chem. In press) and the rotations of asymmetric molecules (Méndez-Fragoso and Ley-Koo, to be submitted), confined in elliptical cones associated with the spheroconal coordinates in which the respective Schrödinger equations are separable, we have recognized the need to use and to construct quasi-periodic Lamé functions. In fact, the new boundary conditions require that the angular momentum label becomes noninteger → λ, and the respective Lamé functions become infinite series. This contribution contains details about the evaluation of the polynomial and quasi-periodic Lamé functions, and their applications in the free particle confined by an elliptical cone with a spherical cap and the harmonic oscillator confined by an elliptical cone.

ABSTRACT:The Schrödinger equation for the Hydrogen atom in spheroconal coordinates admits factorizable solutions, due to the fact that the Hamiltonian, the square of the angular momentum, and a linear combination of the squares of the Cartesian components of the angular momentum form a complete set of commuting operators. The linear combination defines the geometry of a family of elliptical cones, each one of which may serve as a confining boundary for the Hydrogen atom allowing the construction of its exact eigenfunctions expressed as the products of radial Laguerre polynomial functions and quasi-periodic Lamé spheroconal harmonic functions in the elliptical cone coordinates. The boundary condition, requiring the vanishing of the wave function at the confining elliptical cone, introduces noninteger values of the angular momentum quantum label, as well as the breaking of the parity symmetry. Numerical results of the energy spectra, electric dipole moment, and pressure distribution of the Hydrogen atom are reported for different shapes and positions of the confining boundary.

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We study the electronic properties of a hydrogen atom under cylindrical confinement as obtained by a numerical solution to the Schrödinger equation by means of a finite-differences approach. In particular we calculate the dipole oscillator strength, static and dynamic dipole polarizabilities, as well as the mean excitation energy as a function of the position of the hydrogen impurity along the symmetry axis for the case of a 'standard' cylindrical confinement cavity and several confinement conditions. The effect of the displacement on the electronic properties is reflected in the change of the wave-function as the impurity approaches the cylinder potential lid produced by the surrounding confinement environment. We find that the intensity of the main dipole transition,, is reduced as the atom is displaced off-center along the symmetry axis, reaching a minimum half-way between the center of the cylinder and the lid and then increasing when at the cylinder lid. In the process some other transition lines become more intense with a maximum also at half-way between the center and the cylinder lid. We find that the label assignment on the excitation transitions changes as the impurity is displaced along the symmetry axis due to the polarizability of the impurity electronic cloud. Results for the static and dynamic polarizability for the confined impurity as well as the mean excitation energy for the cases of penetrable and impenetrable confinement are presented. We find that the static polarizability increases as the impurity approaches the cylinder lid meanwhile the mean excitation energy is reduced.