Quantum mechanical reactive scattering for three-dimensional atom plus diatom systems. I. Theory

A method is presented for accurately solving the Schrödinger equation for the reactive collision of an atom with a diatomic molecule in three dimensions on a single Born–Oppenheimer potential energy surface. The Schrödinger equation is first expressed in body‐fixed coordinates. The wavefunction is then expanded in a set of vibration–rotation functions, and the resulting coupled equations are integrated in each of the three arrangement channel regions to generate primitive solutions. Next, these are smoothly matched to each other on three matching surfaces which appropriately separate the arrangement channel regions. The resulting matched solutions are linearly combined to generate wavefunctions which satisfy the reactance and scattering matrix boundary conditions, from which the corresponding R and S matrices are obtained. The scattering amplitudes in the helicity representation are easily calculated from the body fixed S matrices, and from these scattering amplitudes several types of differential and integral cross sections are obtained. Simplifications arising from the use of parity symmetry to decouple the coupled‐channel equations, the matching procedures and the asymptotic analysis are discussed in detail. Relations between certain important angular momentum operators in body‐fixed coordinate systems are derived and the asymptotic solutions to the body‐fixed Schrödinger equation are analyzed extensively. Application of this formalism to the three‐dimensional H+H_2 reaction is considered including the use of arrangement channel permutation symmetry, even–odd rotational decoupling and postantisymmetrization. The range of applicability and limitations of the method are discussed

Center-of-Mass Properties of the Exciton in Quantum Wells

We present high-quality numerical calculations of the exciton center-of-mass dispersion for GaAs/AlGaAs quantum wells of widths in the range 2-20 nm. The k.p-coupling of the heavy- and light-hole bands is fully taken into account. An optimized center-of-mass transformation enhances numerical convergence. We derive an easy-to-use semi-analytical expression for the exciton groundstate mass from an ansatz for the exciton wavefunction at finite momentum. It is checked against the numerical results and found to give very good results. We also show multiband calculations of the exciton groundstate dispersion using a finite-differences scheme in real space, which can be applied to rather general heterostructures.Comment: 19 pages, 12 figures included, to be published in Phys. Rev.

The Lazarus project: A pragmatic approach to binary black hole evolutions

We present a detailed description of techniques developed to combine 3D numerical simulations and, subsequently, a single black hole close-limit approximation. This method has made it possible to compute the first complete waveforms covering the post-orbital dynamics of a binary black hole system with the numerical simulation covering the essential non-linear interaction before the close limit becomes applicable for the late time dynamics. To determine when close-limit perturbation theory is applicable we apply a combination of invariant a priori estimates and a posteriori consistency checks of the robustness of our results against exchange of linear and non-linear treatments near the interface. Once the numerically modeled binary system reaches a regime that can be treated as perturbations of the Kerr spacetime, we must approximately relate the numerical coordinates to the perturbative background coordinates. We also perform a rotation of a numerically defined tetrad to asymptotically reproduce the tetrad required in the perturbative treatment. We can then produce numerical Cauchy data for the close-limit evolution in the form of the Weyl scalar $\psi_4$ and its time derivative $\partial_t\psi_4$ with both objects being first order coordinate and tetrad invariant. The Teukolsky equation in Boyer-Lindquist coordinates is adopted to further continue the evolution. To illustrate the application of these techniques we evolve a single Kerr hole and compute the spurious radiation as a measure of the error of the whole procedure. We also briefly discuss the extension of the project to make use of improved full numerical evolutions and outline the approach to a full understanding of astrophysical black hole binary systems which we can now pursue.Comment: New typos found in the version appeared in PRD. (Mostly found and collected by Bernard Kelly

External localization system for mobile robotics

We present a fast and precise vision-based software intended for multiple robot localization. The core component of the proposed localization system is an efficient method for black and white circular pattern detection. The method is robust to variable lighting conditions, achieves sub-pixel precision, and its computational complexity is independent of the processed image size. With off-the-shelf computational equipment and low-cost camera, its core algorithm is able to process hundreds of images per second while tracking hundreds of objects with millimeter precision. We propose a mathematical model of the method that allows to calculate its precision, area of coverage, and processing speed from the camera’s intrinsic parameters and hardware’s processing capacity. The correctness of the presented model and performance of the algorithm in real-world conditions are verified in several experiments. Apart from the method description, we also publish its source code; so, it can be used as an enabling technology for various mobile robotics problems