Exact Solutions of the two-dimensional Schr\"{o}dinger equation with certain central potentials

By applying an ansatz to the eigenfunction, an exact closed form solution of the Schr\"{o}dinger equation in 2D is obtained with the potentials, $V(r)=ar^2+br^4+cr^6$, $V(r)=ar+br^2+cr^{-1}$ and $V(r)=ar^2+br^{-2}+cr^{-4}+dr^{-6}$, respectively. The restrictions on the parameters of the given potential and the angular momentum $m$ are obtained.Comment: Latex files and accepted by Inter. J. Theor. Phys. 39, No.

A new dynamical group approach to the modified Poschl-Teller potential

The properties of the modified P\"{o}schl-Teller (MPT) potential are outlined. The ladder operators are constructed directly from the wave functions without introducing any auxiliary variable. It is shown that these operators are associated to the $su(2)$ algebra. Analytical expressions for the functions $\sinh(\alpha x)$ and $\frac{\cosh(\alpha x)}{\alpha} \frac{d}{dx}$ are evaluated from these ladder operators. The expansions of the coordinate $x$ and momentum $\hat p$ in terms of the $su(2)$ generators are presented. This analysis allows to establish an exact quantum-mechanical connection between the $su(2)$ vibron model and the traditional descriptions of molecular vibron.Comment: Latex file 16 papge

The (2+1) Dirac Equations with δ\delta Potential

In this Letter the bound states of (2+1) Dirac equation with the cylindrically symmetric $\delta (r-r_{0})$-potential are discussed. It is surprisingly found that the relation between the radial functions at two sides of $r_{0}$ can be established by an SO(2) transformation. We obtain a transcendental equation for calculating the energy of the bound state from the matching condition in the configuration space. The condition for existence of bound states is determined by the Sturm-Liouville theorem.Comment: Latex 11 pages accepted by Found. Phys. Let

Exact Solutions to the Schr\"{o}dinger Equation for the potential V(r)=ar2+br−4+cr−6V(r)=a r^2+b r^{-4}+c r^{-6} in 2D

Making use of an ${\it ansatz}$ for the eigenfunctions, we obtain an exact closed form solution to the non-relativistic Schr\"{o}dinger equation with the anharmonic potential, $V(r)=a r^2+b r^{-4}+c r^{-6}$ in two dimensions, where the parameters of the potential $a, b, c$ satisfy some constraints.Comment: Latex file, pages 9 and 2 eps figures, accepted by J. Phys.

Exact Solutions to the Schr\"{o}dinger Equation for the Inverse-Power Potential in Two Dimensions

Utilizing an ${\it ansatz}$ for the eigenfunctions, we arrive at an exact closed form solution to the Schr\"{o}dinger equation with the inverse-power potential, $V(r)=ar^{-4}+br^{-3}+cr^{-2}+dr^{-1}$ in two dimensions, where the parameters of the potential $a, b, c, d$ satisfy a constraint.Comment: Latex file 9 pages and submit to Euro. Phys. J.

Comment on "A new exactly solvable quantum model in NN dimensions" [Phys. Lett. A 375(2011)1431, arXiv:1007.1335]

We pinpoint that the work about "a new exactly solvable quantum model in $N$ dimensions" by Ballesteros et al. [Phys. Lett. A {\bf 375} (2011) 1431, arXiv:1007.1335] is not a new exactly solvable quantum model since the flaw of the position-dependent mass Hamiltonian proposed by them makes it less valuable in physics.Comment: 2 page

Exact solutions to the Dirac equation for a Coulomb potential in D+1D+1 dimensions

The Dirac equation is generalized to $D+1$ space-time.The conserved angular momentum operators and their quantum numbers are discussed. The eigenfunctions of the total angular momenta are calculated for both odd $D$ and even $D$ cases. The radial equations for a spherically symmetric system are derived. The exact solutions for the system with a Coulomb potential are obtained analytically. The energy levels and the corresponding fine structure are also presented.Comment: 12 pages, no figure, RevTex, Accepted by Inter.J.Mod.Phys.

Irreducible bases in icosahedral group space

The irreducible bases in the icosahedral group space are calculated explicitly by reducing the regular representation. The symmetry adapted bases of the system with {\bf I} or {\bf I}$_{h}$ symmetry can be calculated easily and generally by applying those irreducible bases to wavefunctions of the system, if they are not vanishing. As examples, the submatrices of the H\"{u}ckel Hamiltonians for Carbon-60 and Carbon-240 are re-calculated by the irreducible bases.Comment: Revtex 16 page

First principles investigation of nitrogenated holey graphene

Nitrogenated holey graphene (NHG) has attracted much attention because of its semiconducting properties. However, the stacking orders and defect properties have not been investigated. In this letter, the structural and stacking properties of NHG are first investigated. We obtain the most stable stacking structure. Then, the band structures for bulk and multilayer NHG are studied. Impact of the strain on the band gaps and bond characteristics is discuss. In addition, we investigate formation mechanism of native defects of carbon vacancy (VC), carbon interstitial (Ci), nitrogen vacancy (VN), and nitrogen interstitial (Ni) in bulk NHG. Formation energies and transition levels of these native defects are assessed.Comment: 5 pages, 7 figure

Irreducible bases and correlations of spin states for double point groups

In terms of the irreducible bases of the group space of the octahedral double group {\bf O'}, an analytic formula is obtained to combine the spin states $|j,\mu \rangle$ into the symmetrical adapted bases, belonging to a given row of a given irreducible representation of {\bf O'}. This method is effective for all double point groups. However, for the subgroups of {\bf O'}, there is another way to obtain those combinations. As an example, the correlations of spin states for the tetrahedral double group {\bf T'} are calculated explicitly.Comment: Latex file 16 pages,no figur