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

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

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

Semiclassical Casimir Energies at Finite Temperature

We study the dependence on the temperature T of Casimir effects for a range of systems, and in particular for a pair of ideal parallel conducting plates, separated by a vacuum. We study the Helmholtz free energy, combining Matsubara's formalism, in which the temperature appears as a periodic Euclidean fourth dimension of circumference 1/T, with the semiclassical periodic orbital approximation of Gutzwiller. By inspecting the known results for the Casimir energy at T=0 for a rectangular parallelepiped, one is led to guess at the expression for the free energy of two ideal parallel conductors without performing any calculation. The result is a new form for the free energy in terms of the lengths of periodic classical paths on a two-dimensional cylinder section. This expression for the free energy is equivalent to others that have been obtained in the literature. Slightly extending the domain of applicability of Gutzwiller's semiclassical periodic orbit approach, we evaluate the free energy at T>0 in terms of periodic classical paths in a four-dimensional cavity that is the tensor product of the original cavity and a circle. The validity of this approach is at present restricted to particular systems. We also discuss the origin of the classical form of the free energy at high temperatures.Comment: 17 pages, no figures, Late