Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

Silver-based crystalline nanoparticles, microbially fabricated

One mechanism of silver resistance in microorganisms is accumulation of the metal ions in the cell. Here, we report on the phenomenon of biosynthesis of silver-based single crystals with well-defined compositions and shapes, such as equilateral triangles and hexagons, in Pseudomonas stutzeri AG259. The crystals were up to 200 nm in size and were often located at the cell poles. Transmission electron microscopy, quantitative energy-dispersive x-ray analysis, and electron diffraction established that the crystals comprise at least three different types, found both in whole cells and thin sections. These Ag-containing crystals are embedded in the organic matrix of the bacteria. Their possible potential as organic-metal composites in thin film and surface coating technology is discussed.