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Physical Review A

Analytic form for the nonrelativistic Coulomb propagator

S. M. Blinder

Published Jan 1, 1991• 246 citations
Publisher

Abstract

An analytic form for the nonrelativistic Coulomb propagator is derived, thus resolving a long-standing problem in Feynman's path-integral formulation of quantum mechanics. Hostler's formula for the Coulomb Green's function is expanded according to the theorem of Mittag-Leffler, then Fourier transformed term by term to give the Coulomb propagator. The result is a discrete summation over the principal quantum number n, involving Whittaker, Laguerre, Hermite, and error functions. As is the case for other nonquadratic potentials, the Coulomb propagator does not have the canonical structure K=F exp(iS/\ensuremath{\Elzxh}). Part of the expansion resembles a form derived by Crandall [J. Phys. A 16, 3005 (1983)] for the case of reflectionless potentials.

DOI
10.1103/physreva.43.13

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An analytic form for the nonrelativistic Coulomb propagator is derived, thus resolving a long-standing problem in Feynman's path-integral formulation of quantum mechanics. Hostler's formula for the Coulomb Green's function is expanded according to the theorem of Mittag-Leffler, then Fourier transformed term by term to give the Coulomb propagator. The result is a discrete summation over the principal quantum number n, involving Whittaker, Laguerre, Hermite, and error functions. As is the case for other nonquadratic potentials, the Coulomb propagator does not have the canonical structure K=F exp(iS/\ensuremath{\Elzxh}). Part of the expansion resembles a form derived by Crandall [J. Phys. A 16, 3005 (1983)] for the case of reflectionless potentials.
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 K̃ 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.
A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman.The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later.In our approach, with the help of a Laplace transform, a direct way to get the energy dependent Green function is presented, and the propagator can be obtained later with an inverse Laplace transform.The method is illustrated through simple one dimensional examples and for time independent potentials, though it can be generalized to the derivation of more complicated propagators.
The Feynman path integral for regularized nonrelativistic quantum electrodynamics is studied rigorously. We begin with the Lagrangian function of the corresponding classical mechanics and construct the Feynman path integral. In the present paper, the electromagnetic potentials are assumed to be periodic with respect to a large box and quantized through their Fourier coefficients with large wave numbers cut off. Firstly, the Feynman path integral with respect to paths on the space of particles and vector potentials is defined rigorously by means of broken line paths under the constraints. Secondly, the Feynman path integral with respect to paths on the space of particles and electromagnetic potentials is also defined rigorously by means of broken line paths and piecewise constant paths without the constraints. This Feynman path integral is stated heuristically in Feynman and Hibbs' book. Thirdly, the vacuum and the state of photons of given momenta and polarizations are expressed concretely as functions of variables consisting of the Fourier coefficients of vector potentials. It is also proved rigorously in terms of distribution theory that the Coulomb potentials between charged particles naturally appear in the above Feynman path integral approach. This shows that the photons give rise to the Coulomb force.
We solve the Coulomb problem in nonrelativistic quantum mechanics with a charge depending on a parameter that plays the role of time. The choice of this dependence is needed, for example, after certain spatio–temporal transformations when dealing with the interaction of a “small” system (quantum sub-system) with a “large” system, e.g., a bath. These spatio–temporal transformations, combined with path integral, allow us to find the Feynman propagator of the quantum subsystem. To test our approach, we derive the pure Coulomb Green’s function as a limit of our result.