Entangled subspaces and quantum symmetries

Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum j with a spin 1/2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Structure and thermal stability of Langmuir-Blodgett films of barium arachidate

The structures of multilayer Langmuir-Blodgett films of barium arachidate before and after heat treatment have been investigated using both atomic force microscopy (AFM) and grazing incidence synchrotron X-ray diffraction (GIXD). AFM gave information on surface morphology at molecular resolution while GIXD provided quantitative details of the lattice structures of the films with their crystal symmetries and lattice constants. As-prepared films contained three coexisting structures: two triclinic structures with the molecularchains tilted by about 20degrees from the film normal and with 3 x 1 or 2 x 2 super-lattice features arising from height modulation of the molecules in the films; a rectangular structure with molecules perpendicular to the film surface. Of these, the 3 x 1 structure is dominant with a loose correlation between the bilayers. In the film plane both superstructures are commensurate with the local structures, having different oblique symmetries. The lattice constants for the 3 x 1 structure are a(s) = 3a = 13.86 Angstrom, b(s) = b = 4.31 Angstrom and gamma(s) = gamma = 82.7degrees; for the 2 x 2 structure a(s) = 2a = 16.54 Angstrom, b(s) = 2b = 9.67 Angstrom, gamma(s) = gamma = 88degrees. For the rectangular structure the lattice constants are a = 7.39 Angstrom, b = 4.96 Angstrom and gamma = 90degrees. After annealing, the 2 x 2 and rectangular structures were not observed, while the 3 x 1 structure had developed over the entire film. For the annealed films the correlation length in the film plane is about twice that in the unheated films, and in the out-of-plane direction covers two bilayers. The above lattice parameters, determined by GIXD, differed significantly from the values obtained by AFM, due possibly to distortion of the films by the scanning action of the AFM tip. (C) 2004 Published by Elsevier B.V.

Optical trapping of a cube

The successful development and optimisation of optically-driven micromachines will be greatly enhanced by the ability to computationally model the optical forces and torques applied to such devices. In principle, this can be done by calculating the light-scattering properties of such devices. However, while fast methods exist for scattering calculations for spheres and axisymmetric particles, optically-driven micromachines will almost always be more geometrically complex. Fortunately, such micromachines will typically possess a high degree of symmetry, typically discrete rotational symmetry. Many current designs for optically-driven micromachines are also mirror-symmetric about a plane. We show how such symmetries can be used to reduce the computational time required by orders of magnitude. Similar improvements are also possible for other highly-symmetric objects such as crystals. We demonstrate the efficacy of such methods by modelling the optical trapping of a cube, and show that even simple shapes can function as optically-driven micromachines.