82 research outputs found
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 , provided that the dimensionless constant characterizing the
strength of the repulsive potential is large enough, . 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 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 , we find in "closed form" the
polynomial solution P to the associated confluent Heun ODE. We prove that the
Liouvillian solution 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 and , 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
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