© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

International Journal of Quantum Chemistry

Asymptotic forms for the nonrelativistic coulomb propagator

S. M. Blinder

Published Jun 19, 2009• 1 citation

Abstract

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.

DOI: 10.1002/qua.560180807

Related papers

Analytic form for the nonrelativistic Coulomb propagator

S. M. Blinder

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.

Physical Review A•Jan 1, 1991•246 citations

Quantum Coulomb problem in a Gaussian time-dependent electric field within the path integral formalism

Nacira Bedida, Sara Fadhel, Mosbah Difallah +1 more

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.

Theoretical and Mathematical Physics•Apr 1, 2023•0 citations

Green’s function and propagator for the one-dimensionalδ-function potential

S. M. Blinder

A particle in a one-dimensional \ensuremath{\delta}-function potential possesses both discrete and continuum solutions. The configuration-space Green's function and propagator for this problem are derived by explicit summation over the spectrum of eigenstates. The momentum-space Green's function is also obtained. The propagator does not contain the classical action function in any simple way, in contrast to the usual structure in Feynman's path-integral formalism. Various analogies between the \ensuremath{\delta}-function and Coulomb problems are discussed.

Physical Review A•Feb 1, 1988•63 citations

Semiclassical Approximation for the Nonrelativistic Coulomb Propagator

S. M. Blinder

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.

Physical Review Letters•May 14, 1984•12 citations

The Feynman propagator within three parameters generalized Dunkl derivative: One-dimensional case

H. Benzair, Tahar Boudjedaa, M. Merad

In this paper, the Feynman path integral approach within three parameters Dunkl derivative is investigated in the one-dimensional case of nonrelativistic quantum mechanics. We successfully derive the propagator in Cartesian coordinates and give its exact expression for the free particle, inverse square, harmonic oscillator and pure Coulomb potentials. The energy eigenvalues and their corresponding wave functions are determined.

International Journal of Modern Physics A•Oct 30, 2024•2 citations

Feynman-Diagrammatic Description of the Asymptotics of the Time Evolution Operator in Quantum Mechanics

Theo Johnson-Freyd

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on $${\mathbb{R}^n}$$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in $${\hbar}$$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the $${t \to 0}$$ limit approaches the δ distribution. As such, our construction gives explicitly the full $${\hbar\to 0}$$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.

arXiv:1003.1156•Letters in Mathematical Physics•Sep 29, 2010•8 citations

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.

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.

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.

A particle in a one-dimensional \ensuremath{\delta}-function potential possesses both discrete and continuum solutions. The configuration-space Green's function and propagator for this problem are derived by explicit summation over the spectrum of eigenstates. The momentum-space Green's function is also obtained. The propagator does not contain the classical action function in any simple way, in contrast to the usual structure in Feynman's path-integral formalism. Various analogies between the \ensuremath{\delta}-function and Coulomb problems are discussed.

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.

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on

{\mathbb{R}^n}

, with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in

{\hbar}

, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the

{t \to 0}

limit approaches the δ distribution. As such, our construction gives explicitly the full

{\hbar\to 0}

asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.