Oscillation Formula Research Articles

Large-order perturbation expansion of three-dimensional Coulomb systems from four-dimensional anharmonic oscillators.

The three-dimensional Coulomb system can be mapped onto the four-dimensional harmonic oscillator. Additional perturbations of the type ${\mathit{gr}}^{\mathit{p}}$ [r=(${\mathit{x}}^{2}$+${\mathit{y}}^{2}$+${\mathit{z}}^{2}$${)}^{1/2}$] translate into anharmonic perturbations \ensuremath{\lambda}(${\mathbf{x}}^{2}$${)}^{\mathit{p}+1}$ (${\mathbf{x}}^{2}$=${\mathit{x}}_{1}^{2}$+${\mathit{x}}_{2}^{2}$+${\mathit{x}}_{3}^{2}$+${\mathit{x}}_{4}^{2}$) to the oscillator. We use this observation to relate the large-order behavior of perturbation series of the perturbed Coulomb systems to well-known large-order formulas for anharmonic oscillators. In this way, the leading behavior of Coulomb systems can be understood by simple scaling and symmetry arguments within the oscillator systems. Applications to more physical Stark- and Zeeman-type perturbations are briefly discussed.

Large-order perturbation theory of the Zeeman effect in hydrogen from a four-dimensional anisotropic anharmonic oscillator.

The Zeeman Hamiltonian for (spinless) hydrogen in a constant magnetic field is shown to be equivalent to a four-dimensional anisotropic anharmonic oscillator. Using this relation, Rayleigh-Schr\"odinger perturbation series expansions of both systems can be related to each other and analyzed in a unified way. Special emphasis is laid upon analytical estimates of their behavior in large orders of perturbation theory. Employing the path-integral approach, a new large-order formula is derived for the expansion of the ground-state energy of the oscillator system. With use of known Bender-Wu formulas for isotropic anharmonic oscillators, the major part of this calculation becomes straightforward. Combined with the new equivalence, this calculation represents the simplest path-integral derivation of large-order formulas for the Zeeman system.

Infrared reflection and dipole derivatives of azide ion in KN3

The infrared reflection spectrum of the 001 face of tetragonal KN3 has been studied between 100 and 2300 cm−1. Since an oblique incidence technique was employed, observations with TE polarized and TM polarized incident radiation allowed discrimination between crystal modes whose transition dipoles were parallel (A2u) or perpendicular (Eu) to the z axis. The reflection intensity was modeled by a classical harmonic oscillator formula for the dielectric constant, which associates three parameters with each mode. These parameters are νT, the transverse phonon frequency; S, the strength; and γ, the damping constant. From the value of S, taking into account the difference between external and internal effective fields, the values of the dipole derivatives ∂μ/∂q were determined for the several modes. These dipole derivatives were converted to internal coordinates, analyzed from the point of view of the effective charge–charge flux model, and compared with values found for the azide ion in NaN3.

Overlap integral of three-dimensional isotropic harmonic oscillator wave functions

A general formula for the overlap integral of two wave functions of three-dimensional isotropic harmonic oscillators is evaluated in closed form. Four recursion formulas are also derived. The corresponding overlap integral and recursion formulas for one-dimensional oscillators with common origins are obtained in the same form as for the three-dimensional oscillator.