Physical Sciences

Curvature-induced bound-state eigenvalues and eigenfunctions for a particle constrained to move on the surface of a torus are calculated. A limit on the number of bound states that a torus with minor radius a and major radius R can support is obtained. A condition for mapping constrained particle wave functions on the torus into free particle wave functions is established.

The curvature potential arising from confining a particle initially in threedimensional space onto a curved surface is normally derived in the hard constraint q → 0 limit, with q the degree of freedom normal to the surface. In this work the hard constraint is relaxed, and eigenvalues and wave functions are numerically determined for a particle confined to a thin layer in the neighborhood of a toroidal surface. The hard constraint and finite layer (or soft constraint) quantities are comparable, but both differ markedly from those of the corresponding two dimensional system, indicating that the curvature potential continues to influence the dynamics when the particle is confined to a finite layer. This effect is potentially of consequence to the modelling of curved nanostructures.

The Hamiltonian for a particle constrained to move on the surface of a curved nanotube is derived using the methods of differential forms. A two-dimensional Gram-Schmidt orthonormalization procedure is employed to calculate basis functions for determining the eigenvalues and eigenstates of a tubular arc (a nanotube in the shape of a hyperbolic cosine) with several hundred scattering centers. The curvature of the tube is shown to induce bound states that are dependent on the curvature parameter and bend location of the tube. r 2004 Published by Elsevier B.V.

Hadronic form factors for the rare weak transitions Λ b → Λ ( * ) are calculated using a nonrelativistic quark model. The form factors are extracted in two ways. An analytic extraction using single component wave functions (SCA) with the quark current being reduced to its nonrelativistic Pauli form is employed in the first method. In the second method, the form factors are extracted numerically using the full quark model wave function (MCN) with the full relativistic form of the quark current. Although there are differences between the two sets of form factors, both sets satisfy the relationships expected from the heavy quark effective theory (HQET). Differential decay rates, branching ratios and forward-backward asymmetries (FBAs) are calculated for the dileptonic decays Λ b → Λ ( * ) ℓ + ℓ − , for transitions to both ground state and excited daughter baryons. Inclusion of the long distance contributions from charmonium resonances significantly enhances the decay rates. In the MCN model the Λ(1600) mode is the dominant mode in the µ channel when charmonium resonances are considered; the Λ(1520) mode is also found to have a comparable branching ratio to that of the ground state in the µ channel. → N ( * ) were calculated using single component wave functions obtained from a variational diagonalization of a quark model Hamiltonian. In that calculation, quark operators were reduced to their nonrelativistic Pauli form. The decay rates obtained using the form factors extracted in that calculation were in reasonable agreement with experimental results for the semileptonic decays Λ b → Λ c ℓν ℓ and Λ c → Λℓν ℓ .