Analytical Solution Of Dirac Equation Research Articles

Second flavor of heavy ions

The existence of the Second Flavor of Hydrogen Atoms (SFHA), predicted theoretically in 2001, has been confirmed by four different types of atomic or molecular experiments, and evidenced by two different types of astrophysical observations. The SFHA is based on the second analytical solution of the standard Dirac equation. Despite this solution is singular at small r, it can be matched with the regular solution inside the proton for the S-states, and thus becomes legitimate. The SFHA have only the S-states: so, in accordance to the selection rules of quantum mechanics they do not couple to the electromagnetic radiation (except for the 21 cm spectral line): they remain dark. In the present paper we extended the legitimacy of the second analytical solution of the Dirac equation outside the nucleus to heavy ions. We predicted the existence of the second flavor of heavy ions and thus the flavor symmetry of these ions. These ions have only the S-states, so that they remain dark just as the SFHA. This theoretical result has the fundamental importance in its own right. Also, it could encourage experimentalists to perform experiments for the verification of the existence of the second flavor of heavy ions. The most probable candidates are ions whose nuclei are double-magic and thus spherical. As an application of the above fundamental result, we referred to the comparison between the nuclear charge radius rn,μ, determined experimentally by the muonic x-ray transition energies, and the nuclear charge radius rn,e, determined experimentally from the elastic electron scattering. The comparison was possible for two spherical nuclei: for 40Ca20, as well as for 48Ca20. In each case, rn,e turned out to be by about 1% smaller, than rn,μ, the difference being well beyond the experimental error margin. We showed in both above examples, that a relatively small admixture (∼10%) of the second flavor of heavy ions in the target reconciles the values of the nuclear charge radius, measured by two different methods, in favor of the larger value. Finally, we noted that while the SFHA is one of the leading candidates for dark matter or a part of it, there are still unsuccessful attempts to find Weakly Interactive Massive Particles (WIMP) as a part of dark matter. We suggested that if there are massive dark matter particles, then more realistically they could be the second flavor of heavy ions, described in the present paper: they are really dark.

Analytical Solution of Dirac Equation for Hyperbolic Manning-Rosen-like Potential Using Hypergeometric Method for Exact Spin Symmetry

Analytical solution of Dirac equation for hyperbolic Manning-Rosen-like potential is obtained using hypergeometric method. The analytical solution is aimed to determine energy spectrum and radial wave function for this potential with exact spin symmetry. Behavior of atomic particles can be clearly understood if the energy spectrum and wave function of particle are known. Dirac equation for exact spin symmetry is reduced into a second order differential equation like Schrödinger equation. Hypergeometric method for Dirac equation with hyperbolic Manning-Rosen-like potential is the same as for Dirac equation with general hyperbolic Manning-Rosen potential. Energy spectrum is exactly obtained in the closed form and the radial wave functions are expressed in the form of hypergeometric method.

Solution of Dirac equation for modified Poschl Teller plus trigonometric Scarf potential using Romanovsky polynomials method

The approximation analytical solution of Dirac equation for Modified Poschl Teller plus Trigonometric Scarf Potential are investigated numerically in terms of finite Romanovsky Polynomial. The combination of two potentials are substituted into Dirac Equation then the variables are separated into radial and angular parts. The Dirac equation is solved by using Romanovsky Polynomial Method. The equation that can reduce from the second order of differential equation into the differential equation of hypergeometry type by substituted variable method. The energy spectrum is numerically solved using Matlab 2011. Where the increase in the radial quantum number nr and variable of modified Poschl Teller Potential causes the energy to decrease. The radial and the angular part of the wave function also visualized with Matlab 2011. The results show, by the disturbance of a combination between this potential can change the wave function of the radial and angular part.

Analytical Solution of Dirac Equation for q-Deformed Hyperbolic Manning-Rosen Potential in D Dimensions using SUSY QM and its Thermodynamics'Application

The Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.

Pseudospin and spin symmetry of Dirac equation under Deng–Fan potential and Yukawa potential as a tensor interaction

In this paper, we have presented approximately analytical solution of Dirac equation under the pseudospin and spin symmetry for Deng–Fan potential. We have considered Yukawa potential as a tensor interaction. Some numerical results for energy eigenvalues and the corresponding wave functions of Dirac equation in both pseudospin and spin symmetry are reported. The effect of tensor interaction has been also investigated.