Large momentum transfer limit of some matrix elements

The matrix element εfi(K), or ε, that appears in the study of elastic and inelastic electron-atom scattering from an initial state i to a final state f in the first Born approximation depends explicitly on the momentum transfer ℏK⃗ . The uncertainty in the value of the calculated cross sections arises not only from the application of the Born approximation but also from the approximate nature of the wave functions used. For the 1 S1−2 P1 transition in helium, we present an analytic expression in terms of the 1 S1 and 2 P1 wave functions for the leading coefficient C1 in the asymptotic expansion of ε as a power series in 1K; C1 is defined by ε∼C1K5 as K∼∞. An accurate numerical value of C1 is obtained by using a sequence of better and better 1 S1 and 2 P1 wave functions. An accurate value of C1 can be useful in obtaining an approximate analytic form for the matrix element. We also present analytic expressions, in terms of the 1 S1 wave function, for the coefficients of the two leading terms of ε for the diagonal case, that is, for the atomic form factor, and we obtain accurate estimates of those coefficients. The procedure is easily generalizable to other matrix elements of helium, but it would be difficult in practice to apply the procedure to matrix elements of other atoms. We also give a very simple approximate result, valid for a number of matrix elements of heavy atoms, for the ratios of the coefficients of successive terms (in the asymptotically high-K domain) in a power series in 1K. Finally, we plot ε for 1 S1 to 1 S1 and for 1 S1 to 2 P1, with the known low-K and high-K dependence extracted. One might hope that each plot would show little variation, but the 1 S1 to 1 S1 plot varies considerably as one goes to high K, and the 1 S1 to 2 P1 plot shows a very rapid variation for K∼∞, strongly suggesting that at least one element of physics —perhaps a pole outside of but close to the domain of convergence—has been omitted

Van der Waals interactions: Evaluations by use of a statistical mechanical method

In this work the induced van der Waals interaction between a pair of neutral atoms or molecules is considered by use of a statistical mechanical method. Commonly this interaction is obtained by standard quantum mechanical perturbation theory to second order. However, the latter is restricted to electrostatic interactions between charges and dipole moments. So with radiating dipole-dipole interaction where retardation effects are important for large separations of the particles, other methods are needed, and the resulting induced interaction is the Casimir-Polder interaction usually obtained by field theory. It can also be evaluated, however, by a statistical mechanical method that utilizes the path integral representation. We here show explicitly by use of the statistical mechanical method the equivalence of the Casimir-Polder and van der Waals interactions to leading order for short separations where retardation effects can be neglected. Physically this is well known, but in our opinion the mathematics of this transition process is not so obvious. The evaluations needed mean a transform of the statistical mechanical free energy expression to a form that can be identified with second order perturbation theory. In recent works [H{\o}ye 2010] the Casimir-Polder or Casimir energy has been added as a correction to calculations of systems like the electron clouds of molecules. The equivalence to van der Waals interactions to leading order indicates that the added Casimir energy will improve the accuracy of calculated molecular energies. We here also give numerical estimates of this energy including analysis and estimates for the uniform electron gas

Application of an extremum principle to the variational determination of the generalized oscillator strengths of helium

Variational principles have been used extensively for estimating some given functional F(φ, φ†) where the functions φ and φ† are well defined by a set of differential equations and boundary conditions but cannot be determined exactly. The variational principle for the estimation of a matrix element of an arbitrary Hermitian operator W involves not only the trial wave functions φt but also trial auxiliary Lagrange functions Lt; the Lt depend on the φt and on W. To determine the parameters in the Lt efficiently, a functional M(Ltt) is constructed which is an extremum for Ltt=Lt. The technique was recently used successfully in the variational estimation of two diagonal matrix elements. We here use this technique for the variational estimation of an off-diagonal matrix element, the generalized oscillator strengths of helium for the transition between the ground state and the excited 21P state. Two Lt\u27s must be determined. Our results on helium indicate that variational estimates are a significant improvement over the first-order estimates. The results are also compared with those obtained nonvariationally using more elaborate ground-and excited-state wave functions; the comparison represents a check on the method. It is not yet clear which of the two approaches is more efficient

Spatial correlations of vacuum fluctuations and the Casimir-Polder potential

We calculate the Casimir-Polder intermolecular potential using an effective Hamiltonian recently introduced. We show that the potential can be expressed in terms of the dynamical polarizabilities of the two atoms and the equal-time spatial correlation of the electric field in the vacuum state. This gives support to an interesting physical model recently proposed in the literature, where the potential is obtained from the classical interaction between the instantaneous atomic dipoles induced and correlated by the vacuum fluctuations. Also, the results obtained suggest a more general validity of this intuitive model, for example when external boundaries or thermal fields are present.Comment: 7 page

On the lower bound on the exchange-correlation energy in two dimensions

We study the properties of the lower bound on the exchange-correlation energy in two dimensions. First we review the derivation of the bound and show how it can be written in a simple density-functional form. This form allows an explicit determination of the prefactor of the bound and testing its tightness. Next we focus on finite two-dimensional systems and examine how their distance from the bound depends on the system geometry. The results for the high-density limit suggest that a finite system that comes as close as possible to the ultimate bound on the exchange-correlation energy has circular geometry and a weak confining potential with a negative curvature

Universal behavior of dispersion forces between two dielectric plates in the low-temperature limit

The universal analytic expressions in the limit of low temperatures (short separations) are obtained for the free energy, entropy and pressure between the two parallel plates made of any dielectric. The analytical proof of the Nernst heat theorem in the case of dispersion forces acting between dielectrics is provided. This permitted us to formulate the stringent thermodynamical requirement that must be satisfied in all models used in the Casimir physics.Comment: 6 pages, iopart.cls is used, to appear in J. Phys. A (special issue: Proceedings of QFEXT05, Barcelona, Sept. 5-9, 2005

Derivation of the Cubic Non-linear Schr\"odinger Equation from Quantum Dynamics of Many-Body Systems

We prove rigorously that the one-particle density matrix of three dimensional interacting Bose systems with a short-scale repulsive pair interaction converges to the solution of the cubic non-linear Schr\"odinger equation in a suitable scaling limit. The result is extended to $k$-particle density matrices for all positive integer $k$.Comment: 72 pages, 17 figures. Final versio

Diffraction in the Semiclassical Approximation to Feynman's Path Integral Representation of the Green Function

We derive the semiclassical approximation to Feynman's path integral representation of the energy Green function of a massless particle in the shadow region of an ideal obstacle in a medium. The wavelength of the particle is assumed to be comparable to or smaller than any relevant length of the problem. Classical paths with extremal length partially creep along the obstacle and their fluctuations are subject to non-holonomic constraints. If the medium is a vacuum, the asymptotic contribution from a single classical path of overall length L to the energy Green function at energy E is that of a non-relativistic particle of mass E/c^2 moving in the two-dimensional space orthogonal to the classical path for a time \tau=L/c. Dirichlet boundary conditions at the surface of the obstacle constrain the motion of the particle to the exterior half-space and result in an effective time-dependent but spatially constant force that is inversely proportional to the radius of curvature of the classical path. We relate the diffractive, classically forbidden motion in the "creeping" case to the classically allowed motion in the "whispering gallery" case by analytic continuation in the curvature of the classical path. The non-holonomic constraint implies that the surface of the obstacle becomes a zero-dimensional caustic of the particle's motion. We solve this problem for extremal rays with piecewise constant curvature and provide uniform asymptotic expressions that are approximately valid in the penumbra as well as in the deep shadow of a sphere.Comment: 37 pages, 5 figure

A comparison of superradiance and negative-phase-velocity phenomenons in the ergosphere of a rotating black hole

The propagation of electromagnetic plane waves with negative phase velocity (NPV) in the ergosphere of a rotating black hole has recently been reported. A comparison of NPV propagation and superradiance is presented. We show that, although both phenomenons involve negative energy densities, there are two significant differences between them.Comment: Figure 2 in the version published in Phys Lett A is corrected in the arxiv versio

The structure of the atomic helium trimers: Halos and Efimov states

The Faddeev equations for the atomic helium-trimer systems are solved numerically with high accuracy both for the most sophisticated realistic potentials available and for simple phenomenological potentials. An efficient numerical procedure is described. The large-distance asymptotic behavior, crucial for weakly bound three-body systems, is described almost analytically for arbitrary potentials. The Efimov effect is especially considered. The geometric structures of the bound states are quantitatively investigated. The accuracy of the schematic models and previous computations is comparable, i.e. within 20% for the spatially extended states and within 40% for the smaller ^4He-trimer ground state.Comment: 32 pages containing 7 figures and 6 table