Spectra generated by a confined softcore Coulomb potential

Analytic and approximate solutions for the energy eigenvalues generated by a confined softcore Coulomb potentials of the form a/(r+\beta) in d>1 dimensions are constructed. The confinement is effected by linear and harmonic-oscillator potential terms, and also through `hard confinement' by means of an impenetrable spherical box. A byproduct of this work is the construction of polynomial solutions for a number of linear differential equations with polynomial coefficients, along with the necessary and sufficient conditions for the existence of such solutions. Very accurate approximate solutions for the general problem with arbitrary potential parameters are found by use of the asymptotic iteration method.Comment: 17 pages, 2 figure

Dirac equation exact solutions for generalized asymmetrical Hartmann potentials

In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of vector and scalar generalized Hartmann potentials. This is done provided the vector potential is equal to or minus the scalar potential. The cases of some quasi-exactly solvable and Morse-like potentials are briefly commented.Comment: 8 pages, no figure

Classical Monopoles: Newton, NUT-space, gravomagnetic lensing and atomic spectra

Stimulated by a scholium in Newton's Principia we find some beautiful results in classical mechanics which can be interpreted in terms of the orbits in the field of a mass endowed with a gravomagnetic monopole. All the orbits lie on cones! When the cones are slit open and flattened the orbits are exactly the ellipses and hyperbolae that one would have obtained without the gravomagnetic monopole. The beauty and simplicity of these results has led us to explore the similar problems in Atomic Physics when the nuclei have an added Dirac magnetic monopole. These problems have been explored by others and we sketch the derivations and give details of the predicted spectrum of monopolar hydrogen. Finally we return to gravomagnetic monopoles in general relativity. We explain why NUT space has a non-spherical metric although NUT space itself is the spherical space-time of a mass with a gravomagnetic monopole. We demonstrate that all geodesics in NUT space lie on cones and use this result to study the gravitational lensing by bodies with gravomagnetic monopoles. We remark that just as electromagnetism would have to be extended beyond Maxwell's equations to allow for magnetic monopoles and their currents so general relativity would have to be extended to allow torsion for general distributions of gravomagnetic monopoles and their currents. Of course if monopoles were never discovered then it would be a triumph for both Maxwellian Electromagnetism and General Relativity as they stand!Comment: 39 pages, 9 figures and 2 tables available on request from the author

Quantum singular oscillator as a model of two-ion trap: an amplification of transition probabilities due to small time variations of the binding potential

Following the paper by M. Combescure [Ann. Phys. (NY) 204, 113 (1990)], we apply the quantum singular time dependent oscillator model to describe the relative one dimensional motion of two ions in a trap. We argue that the model can be justified for low energy excited states with the quantum numbers $n\ll n_{max}\sim 100$, provided that the dimensionless constant characterizing the strength of the repulsive potential is large enough, $g_*\sim 10^5$. Time dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator, and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center of mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential, since the large value of $g_*$ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.Comment: 19 pages, LaTeX, 5 eps-figures, to appear on Phys. Rev. A, one reference correcte

On the Liouvillian solutions to the perturbation equations of the Schwarzschild black hole

We use Kovacic's algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of the master equation which governs the evolution of first order perturbations of the Schwarzschild geometry. We show that all solutions in quadratures of this equation contain a polynomial solution to an associated ordinary differential equation (ODE). This ODE, apart from a few trivial cases, falls into the confluent Heun class. In the case of the gravitational perturbations, for the Liouvillian solution $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$, we find in "closed form" the polynomial solution P to the associated confluent Heun ODE. We prove that the Liouvillian solution $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$ is a product of elementary functions, one of them being the polynomial P. We extend previous results by Hautot and use the extended results we derive in order to prove that P admits a finite expansion in terms of truncated confluent hypergeometric functions of the first kind. We also prove, by using the extended results we derive, that P admits also a finite expansion in terms of associated Laguerre polynomials. We prove, save for two unresolved cases, that the Liouvillian solutions $\chi$ and $\chi \int \frac {{\rm d}r_{\!\ast}}{\chi^{2}}$, initially found by Chandrasekhar, are the only Liouvillian solutions to the master equation. We improve previous results in the literature on this problem and compare our results with theirs. Comments are made for a more efficient implementation of Kovacic's algorithm to any second order ODE with rational function coefficients. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher.Comment: 118 page

La structure des raies K des atomes très légers - (Deuxième article)

Dans un article publié ici même (A. HAUTOT, J.Phys. Radium (mai 1933), t. 4, p. 236), j'ai décrit la structure fine des raies K du carbone et du bore et j'ai donné quelques indications au sujet des appareils qui m'ont permis de mettre cette structure fine en évidence. Le présent article apporte quelques précisions nouvelles au sujet de ces appareils ainsi que des résultats expérimentaux nouveaux relatifs aux raies K de l'oxygène, de l'azote, du carbone, du bore et du glucinium ; ainsi se poursuit l'étude à grande dispersion de la structure des raies K appartenant à la première rangée horizontale du système périodique des éléments. Ensuite, je rappelle la théorie de Langer qui donne l'interprétation correcte des satellites des spectres de rayons X des atomes moyens et lourds et qui parait devoir s'appliquer encore aux éléments de la première rangée horizontale du système périodique à partir du carbone. Le rayonnement K anormal du bore etdu glucinium ayant été attribué par certains chercheurs à l'existence, dans le cristal, d'électrons de valence à l'état libre ou plus ou moins lié, j'apporte des faits expérimentaux en désaccord avec ce point de vue; il semble plutôt qu'il faille attribuer ce rayonnement anormal à l'existence, dans les atomes de bore et de glucinium, de passages défendus non quantifiés