Exact solution of Schrödinger equation for Pseudoharmonic potential

Exact solution of Schr?dinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of ℓ and n with n ≤ 5 for some diatomic molecules.      

Ultrastructural description of spermiogenesis within the Mediterranean Gecko, Hemidactylus turcicus (Squamata: Gekkonidae)

We studied spermiogenesis in the Mediterranean Gecko, Hemidactylus turcicus, at the electron microscope level and compared to what is known within other Lepidosaurs. In H. turcicus germ cells are connected via cytoplasmic bridges where organelle and cytoplasm sharing is observed. The acrosome develops from merging transport vesicles that arise from the Golgi and subsequently partition into an acrosomal cap containing an acrosomal cortex, acrosomal medulla, perforatorium, and subacrosomal cone. Condensation of DNA occurs in a spiral fashion and elongation is aided by microtubules of the manchette. A nuclear rostrum extends into the subacrosomal cone and is capped by an epinuclear lucent zone. Mitochondria and rough endoplasmic reticulum migrate to the posterior portion of the developing germ cell during the cytoplasmic shift and the flagellum elongates. Mitochondria surround the midpiece as the anlage of the annulus forms. The fibrous sheath begins at mitochondrial tier 3 and continues into the principal piece. Peripheral fibers associated with microtubule doublets 3 and 8 are grossly enlarged. During the final stages of germ cell development spermatids are wrapped with a series of Sertoli cell processes, which exhibit ectoplasmic specializations and differing cytoplasmic consistencies. The results observed here corroborate previous studies, which show the conservative nature of sperm morphology. However, ultrastructural character combinations specific to sperm and spermiogenesis seem to differ among taxa. Further studies into sperm morphology are needed in order to judge the relevance of the ontogenic changes recorded here and to determine their role in future studies on amniote evolution.     

Approximate analytical solutions of the effective mass Dirac equation for the generalized Hulthén potential with any <Emphasis Type="Italic">κ</Emphasis>-value

The Dirac equation, with position-dependent mass, is solved approximately for the generalized Hulthén potential with any spin-orbit quantum number κ. Solutions are obtained by using an appropriate coordinate transformation, reducing the effective mass Dirac equation to a Schrödinger-like differential equation. The Nikiforov-Uvarov method is used in the calculations to obtain energy eigenvalues and the corresponding wave functions. Numerical results are compared with those given in the literature. Analytical results are also obtained for the case of constant mass and the results are in good agreement with the literature.     

Approximate analytical solutions of the generalized Woods-Saxon potentials including the spin-orbit coupling term and spin symmetry

We study the approximate analytical solutions of the Dirac equation for the generalized Woods-Saxon potential with the pseudo-centrifugal term. We apply the Nikiforov-Uvarov method (which solves a second-order linear differential equation by reducing it to a generalized hypergeometric form) to spin- and pseudospin-symmetry to obtain, in closed form, the approximately analytical bound state energy eigenvalues and the corresponding upper- and lower-spinor components of two Dirac particles. The special cases κ = ±1 (s = $ \tilde l $ = 0, s-wave) and the non-relativistic limit can be reached easily and directly for the generalized and standard Woods-Saxon potentials. We compare the non-relativistic results with those obtained by others.     

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

A new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form $ V(r,\theta ) = \tfrac{1} {8}\kappa r_e^2 \left( {\tfrac{r} {{r_e }} - \tfrac{{r_e }} {r}} \right)^2 + \tfrac{{\beta cos^2 \theta }} {{r^2 sin^2 \theta }} A new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form . The energy eigenvalues and eigenfunctions of the bound-states for the Schr?dinger equation in D-dimensions for this potential are obtained analytically by using the Nikiforov-Uvarov method. The radial and angular parts of the wave functions are obtained in terms of orthogonal Laguerre and Jacobi polynomials. We also find that the energy of the particle and the wave functions reduce to the energy and the wave functions of the bound-states in three dimensions.      

On a superadditivity property of Gram’s determinant

Summary Some superadditivity properties are given for Gram determinants involving inner products on a linear space.     

Hyperbolic systems of conservation laws with some special invariance properties

The Euler equations of fluid dynamics are an example of a very special class of nonlinear, hyperbolic systems of conservation laws, in particular those satisfying conditions of reflection and Galilean invariance. These invariance properties are directly responsible for several of the attractive structural features of this system.     

Bound states of a more general exponential screened Coulomb potential

An alternative approximation scheme has been used in solving the Schr?dinger equation to the more general case of exponential screened Coulomb potential, V(r) = −(a/r)[1 + (1 + br)e ^−2br ]. The bound state energies of the 1s, 2s and 3s-states, together with the ground state wave function are obtained analytically upto the second perturbation term.      

Supersymmetric solutions of <Emphasis Type="Italic">PT</Emphasis>-/non-<Emphasis Type="Italic">PT</Emphasis>-symmetric and non-Hermitian screened Coulomb potential via Hamiltonian hierarchy inspired variational method

The supersymmetric solutions of PT -symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schr?dinger equation for the Exponential-Cosine Screened Coulomb potential. The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.