Nonrelativistic Coulomb Green Research Articles

Asymptotic forms for the nonrelativistic coulomb propagator

The nonrelativistic Coulomb Green's function G(r1,r2,k) = G(x,y,k) (where x ≡ r1 + r2 + r12, y ≡ r1 + r2 − r12) was worked out in closed form by Hostler in 1963. There is as yet no closed form for the corresponding propagator K(x,y,t). The unavailability of this function is a serious defect of Feynman's path-integral formulation of quantum mechanics. It has moreover created a bottleneck in our formalism for generating atomic and molecular eigenvalue spectra by time-dependent Green's function techniques. As, perhaps, a first step toward the ultimate realization of a closed form, we have worked out asymptotic forms for K in the limiting domains as Z 0, t 0, and r ∞. We have exploited the structure of the propagator, whereby K = F exp (iS), in terms of the classical action integral S. Our main result is a first-order approximation to the Coulomb propagator, which can be connected to a corresponding approximation for G. We also propose a simpler function K for possible application in the aufbau of many-electron Green's functions. As such, this function might play a role analogous to that of Slater-type orbitals in conventional quantum chemistry.

Dynamic toroid polarizability of atomic hydrogen

The concept of toroid polarizability introduced in previous work is examined within the familiar context of nonrelativistic quantum Coulomb systems, in a way unbiased by approximations. The dynamic (i.e., frequency (ω) dependent) toroid dipole polarizability γ(ω) of a (nonrelativistic, spinless, ground-state) hydrogenlike atom is calculated analytically in terms of (essentially) one Gauss hypergeometric function. The static result takes on the simple form γ(ω = 0) = 23 60 α 2Z −4a 0 5 (α = fine structure constant, Z = nucleus charge number, a 0 = Bohr radius). γ(ω) characterizes the linear response of the system to a conduction and/or displacement (time dependent) external current. The method of calculation (based on the use of the integral representation for the nonrelativistic Coulomb Green's function) is presented in detail. The imaginary part of γ(ω) above ionization threshold is also computed in a simple closed form. Comparing γ(ω = 0) with the already known (exact) results for the electric multipole polarizabilities (for which, as a byproduct, we present in an Appendix a considerably simplified expression, to our knowledge the simplest reported as yet), one sees that although for H-like atoms the toroid effects appear as very small indeed, they are however increasing with Z. A comparison with analogous (but, this time, only order of magnitude) evaluations for (charged) pions, indicates that the role of the induced toroid moments (as against that of the usual electric ones) increases drastically when passing from atomic to hadron physics; it is argued that this trend might continue further, at the sub-hadronic level.

Dynamic toroid polarizability of atomic hydrogen: A. Costescu. Department of Physics, University of Bucharest, P.O. Box Mg-11, Bucharest, Romania; and E. E. Radescu. International Center for Theoretical Physics, Trieste, Italy, and Institute of Atomic Physics, P.O. Box Mg-6, Bucharest, Romania

Abstract The concept of toroid polarizability introduced in previous work is examined within the familiar context of nonrelativistic quantum Coulomb systems, in a way unbiased by approximations. The dynamic (i.e., frequency (ω) dependent) toroid dipole polarizability γ(ω) of a (nonrelativistic, spinless, ground-state) hydrogenlike atom is calculated analytically in terms of (essentially) one Gauss hypergeometric function. The static result takes on the simple form γ(ω = 0) = 23 60 α 2 Z −4 a 0 5 (α = fine structure constant, Z = nucleus charge number, a0 = Bohr radius). γ(ω) characterizes the linear response of the system to a conduction and/or displacement (time dependent) external current. The method of calculation (based on the use of the integral representation for the nonrelativistic Coulomb Green's function) is presented in detail. The imaginary part of γ(ω) above ionization threshold is also computed in a simple closed form. Comparing γ(ω = 0) with the already known (exact) results for the electric multipole polarizabilities (for which, as a byproduct, we present in an Appendix a considerably simplified expression, to our knowledge the simplest reported as yet), one sees that although for H-like atoms the toroid effects appear as very small indeed, they are however increasing with Z. A comparison with analogous (but, this time, only order of magnitude) evaluations for (charged) pions, indicates that the role of the induced toroid moments (as against that of the usual electric ones) increases drastically when passing from atomic to hadron physics; it is argued that this trend might continue further, at the sub-hadronic level.

Double laplace transforms of the radial nonrelativistic Coulomb Green function

General double Laplace transforms of the radial nonrelativistic Coulomb Green function are calculated and presented in several different forms. A special case, known from previous studies is found to be consistent with this work. Various representations of the Green function are utilized, of which some are apparently new. In addition, a generalization of the double Laplace transform involving two spherical Bessel functions, is calculated. These transforms are of interest since they appear in many atomic calculations, particularly those including retardation effects.

Nonrelativistic Coulomb Green’s function in parabolic coordinates

The nonrelativistic Coulomb Green’s function G(+)(r1,r2,k) is evaluated by explicit summation over discrete and continuum eigenstates in parabolic coordinates. This completes the derivation of Meixner, who was able to obtain only the r1=0 and r2→∞ limiting forms of the Green’s function. Further progress is made possible by an integral representation for a product of two Whittaker functions given by Buchholz. We obtain the closed form for the Coulomb Green’s function previously derived by Hostler, via an analogous summation in spherical polar coordinates. The Rutherford scattering limit of the Green’s function is also demonstrated, starting with an integral representation in parabolic coordinates.

Algebraic Equations for Bethe-Salpeter and Coulomb Green's Functions

The equation for the Bethe-Salpeter Green's function in the case of two scalar quarks interacting via the exchange of a scalar particle of zero mass is transformed to an algebraic equation with the help of the dynamical group SO(5, 2). From this equation a one-parameter integral representation of the Green's function is obtained in the case of maximal binding, and from this representation the Green's function is calculated in terms of a hypergeometric function. The equation for the f-dimensional nonrelativistic Coulomb Green's function is also transformed to an algebraic equation.

Remarks on the Construction of the Nonrelativistic Coulomb Green's Function

Remarks on the Construction of the Nonrelativistic Coulomb Green's Function

Excited State Reduced Coulomb Green's Function and f-Dimensional Coulomb Green's Function in Momentum Space

An integral representation for the f-dimensional nonrelativistic Coulomb Green's function in momentum space and an expansion of this function in a series of Gegenbauer polynomials are obtained. It is shown that the momentum space representatives of the f-dimensional Coulomb Green's function and the related reduced Green's functions can be obtained by differentiation with respect to the momentum transfer of the corresponding functions in the one-dimensional (f odd) or two-dimensional (f even) case. Expressions in closed form are then obtained for the momentum space representatives in the one-dimensional case of the full Coulomb Green's function and of the general nth excited state reduced Coulomb Green's function.

Coulomb Green's Function in f-Dimensional Space

It is shown that the f-dimensional nonrelativistic Coulomb Green's function and the associated reduced Green's functions can be obtained by differentiation of the corresponding functions in the 1-dimensional (f odd) or 2-dimensional (f even) case. A new expansion of the 3-dimensional coordinate-space Coulomb Green's function and a new sum formula for a product of two Laguerre polynomials with different arguments are derived.

Nonrelativistic Coulomb Green's Function in Momentum Space

The nonrelativistic Coulomb Green's function in momentum space is obtained in closed form by Fourier transforming the known expression for the coordinate-space Green's function. Also, an integral representation for the momentum-space Green's function is obtained which looks rather attractive from the point of view of applications.

Representations for the Nonrelativistic Coulomb Green's Function

Derivation of the representations for the nonrelativistic Coulomb Green's function is discussed. It is shown that the recently published representations, both in integral and closed forms, can be obtained from an expression for the difference of the diverging and converging wave solutions of the Green's function which is given in terms of the wavefunctions summed by Gordon.

Eigenfunction Form of the Nonrelativistic Coulomb Green's Function

By treating the contour integrals in the momentum plane, the conventional eigenfunction form is deduced from the characteristic form for the nonrelativistic Coulomb Green's function. A straight-forward extension of the argument employed here leads to the most general form of the eigenfunction form.

Integral Representation for the Nonrelativistic Coulomb Green's Function

Although the radial Green's function for the Schrödinger equation in a Coulomb field can be obtained in the usual way in terms of the two linearly independent solutions to the radial equation for a particular angular momentum state, the sum over angular momentum states does not seem to have been carried out. In this note this sum is carried out and a ``closed form'' obtained in the form of a double integral. The result is believed to be useful for perturbation calculations where the ``intermediate states'' involve many angular momentum states.