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Physical Review Letters

Semiclassical Approximation for the Nonrelativistic Coulomb Propagator

S. M. Blinder

Published May 14, 1984• 12 citations
Publisher

Abstract

An approximation to the Coulomb propagator, correct to first order in $\ensuremath{\hbar}$, is derived. This function has the structure $K=F(\ensuremath{\lambda},\ensuremath{\mu},\ensuremath{\nu})\mathrm{exp}[iS(\ensuremath{\lambda},\ensuremath{\mu},\ensuremath{\nu})]$, in terms of auxiliary variables $\ensuremath{\lambda}$, $\ensuremath{\mu}$, $\ensuremath{\nu}$ introduced in the solution of the corresponding Hamilton-Jacobi equation.

DOI
10.1103/physrevlett.52.1771

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An approximation to the Coulomb propagator, correct to first order in \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ, is derived. This function has the structure K=F(\ensuremathλ,\ensuremathμ,\ensuremathν)exp[iS(\ensuremathλ,\ensuremathμ,\ensuremathν)]K=F(\ensuremath{\lambda},\ensuremath{\mu},\ensuremath{\nu})\mathrm{exp}[iS(\ensuremath{\lambda},\ensuremath{\mu},\ensuremath{\nu})]K=F(\ensuremathλ,\ensuremathμ,\ensuremathν)exp[iS(\ensuremathλ,\ensuremathμ,\ensuremathν)], in terms of auxiliary variables \ensuremathλ\ensuremath{\lambda}\ensuremathλ, \ensuremathμ\ensuremath{\mu}\ensuremathμ, \ensuremathν\ensuremath{\nu}\ensuremathν introduced in the solution of the corresponding Hamilton-Jacobi equation.
We derive a regularized semiclassical radial propagator for the Coulomb potential, a case for which standard approaches run into well-known difficulties associated with a non-Cartesian radial coordinate and a potential singularity. Following Kleinert [Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific, Singapore, 1990)], we first perform a quantum-mechanical regularization of the propagator. The semiclassical limit is then obtained by stationary phase approximation of the resulting integrals. The semiclassical propagator so derived has the standard Van Vleck--Gutzwiller form for the radial Coulomb problem with a potential correction (Langer modification) term included. The regularized semiclassical propagator is applied to compute the autocorrelation function for a Gaussian Rydberg wave packet.
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.
This work explores the quantum propagator K(x,t)K(x,t)K(x,t) as a solution of the system's dynamical equation. We develop a generalized propagator framework in which the propagator is written in the form K=exp⁡ ⁣(R+iℏS)K=\exp\!\left(R+\frac{i}{\hbar}S\right)K=exp(R+ℏi​S), where SSS governs the semiclassical phase structure and RRR governs the amplitude transport and weighting of configurations. Starting from nonrelativistic quantum mechanics, the classical Hamilton--Jacobi equation emerges in the semiclassical limit, while RRR reduces to the logarithm of the Van Vleck determinant. The formalism is then extended to relativistic field theory and minisuperspace quantum cosmology using functional methods and generalized Hamiltonian constraints. Then the resulting semiclassical equations recover the corresponding classical dynamics along characteristic flows. In minisuperspace models, the formalism yields coupled equations for geometric and matter sectors analogous to semiclassical Wheeler--DeWitt systems. In semiclassical gravitational settings, the gravitational contribution may admit an entropy-like interpretation, consistent with thermodynamic weighting factors appearing in Euclidean gravitational path integrals. The resulting framework suggests a unified semiclassical description in which propagator phases encode dynamical evolution while amplitude functionals encode transport and configuration weighting. This perspective motivates viewing mechanics as a relation between the differential structure of spacetime and the differential structure of the space of dynamical configurations.