Coulomb Green Function Research Articles

The Coulomb-Green's function calculation of the second-order interaction of a hydrogen atom with the radiation field

The sums over hydrogen spectrum, describing the second-order interaction of a hydrogen atom with the radiation field, are evaluated by means of the Coulomb-Green's function. These sums are expressed in terms of well-known functions. The results are applied to the ground-state dynamical polarizability and two-photon decay of the metastable 2 1S 1 2 state.

Three-Body Problem with Charged Particles

The formal modification, by long-range potentials, of the scattering equations describing three particles interacting through short-range potentials is discussed in terms of the exact Coulomb Green's function, whose mathematical properties are known. This Green's function can be explicitly written down in several special cases; approximate forms of the modified theory are described for these instances. The application of the Coulomb-modified three-particle scattering theory to the $\ensuremath{\alpha}$-cluster model of ${\mathrm{C}}^{12}$ and to deuteron-induced nuclear reactions is described in some detail. For the three-$\ensuremath{\alpha}$ model, an heuristic generalization of the local plane-wave approximation is used to obtain a simple approximate form for the Coulomb Green's function. This new theory may have some relevance to the problem of helium burning in stars. Finally, in the Appendix a discussion is given of why the methods of multiparticle collision theory fail in the problem of calculating the general three-body Coulomb Green's function at positive, real energies. A possibly useful, mathematically well-defined method for constructing this Green's function at negative real energies is also given in the Appendix.

Runge-Lenz Vector and the Coulomb Green's Function

The fact that the Runge-Lenz vector is an extra constant of the motion for a charged particle moving in a Coulomb potential is found to account for the especially simple structure of the nonrelativistic coordinate space Coulomb Green's function. Also, the study of the consequences of this extra constant of the motion leads to a separation of variables in the differential equation for the coordinate space Coulomb Green's function, and hence to a new derivation of the closed-form expression for this Green's function which avoids the use of infinite series and the detailed properties of special functions.

Coulomb Green's Function

A one-parameter integral representation is given for the momentum space Green's function of the nonrelativistic Coulomb problem.

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.

Coulomb Green's Functions and the Furry Approximation

The Coulomb Green's function for the nonrelativistic Schrödinger equation is obtained in closed form starting from the partial-wave expansion and using an integral representation for a product of two Whittaker functions with different arguments. The Neumann's series for Jv(kz) is required in evaluating the sum on states. Using the same methods, the Coulomb Green's functions for the Klein-Gordon and iterated Dirac equations are obtained in closed form in the ``Furry approximation,'' a2/(J + ½)2 ≪ 1, a = Ze2/4πh/c. The Klein-Gordon Green's function in this approximation is shown to be at the same time the exact Green's function for the Klein-Gordon equation without the potential squared term. An alternate and very simple derivation of the approximate Green's function for the iterated Dirac equation is given using perturbation theory. From this Green's function, an approximate Coulomb Green's function in closed form for the Dirac equation itself is constructed. Certain known results for Coulomb wavefunctions with modified plane-wave behavior at large distances are rederived using the foregoing methods and results.

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.

Relativistic coulomb-like green's functions in closed form

It is pointed out that an improvement can be made over approximate solutions of Coulomb Green's functions. The relativistic Green's fuaction is obtained to an exact factorization of the iterated Dirac-Coulomb Hamiltonian with the single term ( alpha Z/r)/sup 2/ removed. (R.E.U.)

Coulomb Green's Function in Closed Form

A simple derivation of the nonrelativistic Green's function is given. Other representations of the function and their agreement with this and various other representations are discussed. (D.C.W.)

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.

Characteristic Coulomb Green's Function and Its Eigenfunction Expansion

The eigenfunction form of the Green's function is derived from the characteristic form of the Green's function for the nonrelativistic Coulomb operator. This derivation shows how the wave boundary condition of the characteristic form of the Green's function is related to the branch cut in the continuous part of the spectrum. An application of the eigenfunction expansion form of this Green's function is discussed.

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.