Quantum Inequalities in Quantum Mechanics

We study a phenomenon occurring in various areas of quantum physics, in which an observable density (such as an energy density) which is classically pointwise non-negative may assume arbitrarily negative expectation values after quantization, even though the spatially integrated density remains non-negative. Two prominent examples which have previously been studied are the energy density (in quantum field theory) and the probability flux of rightwards-moving particles (in quantum mechanics). However, in the quantum field context, it has been shown that the magnitude and space-time extension of negative energy densities are not arbitrary, but restricted by relations which have come to be known as quantum inequalities. In the present work, we explore the extent to which such quantum inequalities hold for typical quantum mechanical systems. We derive quantum inequalities of two types. The first are kinematical quantum inequalities where spatially averaged densities are shown to be bounded below. Specifically, we obtain such kinematical quantum inequalities for the current density in one spatial dimension (imposing constraints on the backflow phenomenon) and for the densities arising in Weyl–Wigner quantization. The latter quantum inequalities are direct consequences of sharp Gårding inequalities. The second type are dynamical quantum inequalities where one obtains bounds from below on temporally averaged densities. We derive such quantum inequalities in the case of the energy density in general quantum mechanical systems having suitable decay properties on the negative spectral axis of the total energy.Furthermore, we obtain explicit numerical values for the quantum inequalities on the one-dimensional current density, using various spatial averaging weight functions. We also improve the numerical value of the related backflow constant previously investigated by Bracken and Melloy. In many cases our numerical results are controlled by rigorous error estimates.submitted 27/01/04, accepted 05/05/04     

Second harmonic generation with surface plasmons in alkali metals

We calculate the reflected second harmonic light from alkali metal films with the simultaneous excitation of the surface plasmon mode. The harmonic generation from a sodium film at the ruby laser frequency increases by over two orders of magnitude at the angle for surface plasmon excitation. The harmonic enhancement is closely related to the surface plasmon density and exhibits a strong dependence on the angle of incidence, film thickness, and the linear optical constants of the metal film.     

Optical second harmonic generation with surface plasmons in piezoelectric crystals

We report the first observation of reflected optical second harmonic generation of light due to excitation of fundamental and harmonic surface plasmon modes at the interface between a metallic film and a piezoelectric crystal. Excitation of the fundamental surface plasmon mode produces a large enhancement of the piezoelectric generated reflected harmonic light whereas excitation of the nonlinear mode produces only a small minimum in the background surface generated harmonic light.     

Die KristallStruktur von LiAlCl4 · 3 SO2

Crystal Structure of LiAlCl₄ · 3 SO₂ Single crystals of LiAlCl₄ · 3SO₂ were grown on a 4-circle diffractometer in a temperature gradient 17°/40°C and diffraction data were collected at ?10°C. The compound crystallizes in the orthorhombic system, space group Pnam (No. 62) with a = 951.6(5), b = 1327.1(9), c = 1017.4(8) pm, Z = 4. The atomic parameters were refined to R = 0.049. The characteristic structural units are parallel strings of the composition Li(SO₂)_6/2. The spaces between these strings are occupied by nearly ideal tetrahedral AlCl₄ ions. The close relationship with the structure of LiClO₄ ·% 3H₂O is discussed.     

Characterization and comparison of soluble and immobilized pig muscle aldolases

Pig muscle aldolase was insolubilized by covalent attachment to a polyacrylamide matrix containing carboxylic functional groups. The catalytic activity of the Akrilex C-aldolase was 2014 units/g solid, i.e., an activity loss of only about 5% relative to the initial activity. The pH optimum for catalytic activity shifted form 7.25 to 7.5 and the apparent temperature optimum from 313 to 318 K. The Michaelis constant of the insolubilized enzyme was significantly higher than that of the soluble aldolase. Heat- and urea-inactivation experiments revealed that the immobilization increased the stability of the enzyme.     

Characterizations of locally testable events

Let Σ be a finite alphabet, Σ^* the free monoid generated by Σ and χ the length of χ ∈ Σ^*. For any integer k0, f_k(χ) (t_k(χ)) is χ if χ < k + 1, and it is the prefix (suffix) of χ of length k, othewise. Also let m_k+1(χ) = {νχ = uνw and ν = k+1}. For χ, y ε Σ^* define χ ～ _k+1y iff f_k(χ) = f_k(y), t_k(χ) = t_k(y) and m_k+1(χ) = m_k+1(y). The relation ～_k+1 is a congruence of finite index over Σ^*. An event E ? Σ^* is (k+1)-testable iff it is a union of congruence classes of ～_k+1. E is locally testable (LT) if it is k+1-testable for some k. (This definition differs from that of [6] but is equivalent.)We show that the family of LT events is a proper sub-family of star-free events of dot-depth 1. LT events and k-testable events are characterized in terms of (a) restricted star-free expressions based on finite and cofinite events; (b) finite automata accepting these events; (c) semigroups; and (d) structural decomposition of such automata. Algorithms are given for deciding whether a regular event is (a) LT and (b) k+1-testable. Generalized definite events are also characterized.     

Consensus of opinion

In a decision analysis, it is often necessary to combine a group of individuals' beliefs into a consensus probability distribution. This paper addresses the question whether it is possible to base such consensus distributions only upon the information present within the group or must some arbitrary rule be used to resolve disagreement. Some earlier work on modifying beliefs in the light of another's opinion is developed to apply to groups of n people. Using this as a “benchmark of rationality”, standard methods of forming group consensus probability distributions are found somewhat arbitrary. Furthermore it is argued that the possibility of constructing better procedures is remote.