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Theoretical and Mathematical Physics

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

Nacira Bedida, Sara Fadhel, Mosbah Difallah, Mohammed Tayeb Meftah

Published Apr 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s0040577923040062

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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 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.
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 present an exact analytical solution for the quantum dynamics of a charged particle subjected to both a time-dependent electric field and a static magnetic field aligned along the zzz -direction. Using a systematic approach based on successive unitary transformations, we reduce the original three-dimensional problem to a two-dimensional system of decoupled, time-dependent harmonic oscillators. This technique produces free parameters that allow us to impose constrains to derive the exact solution of Schrödinger equation with the time-dependent Hamiltonian through the explicit derivation of the quantum propagator and shows the equivalence of this approach to established path integral methods for such systems. The developed framework provides new insights into quantum systems with time-dependent electromagnetic fields and offers analytical solutions.
In a reasonably self-contained presentation mathematical rigour is supplied to the important ideas of solving certain non-Gaussian path integrals by changes of dimension and/or path-dependent time transformations. The resulting genuine path-integral calculus neither requires discretization prescriptions nor sophisticated methods from the theory of stochastic differential equations. The power of the calculus is illustrated by two standard quantum-physics applications. First, the calculation of the time-dependent propagator corresponding to a particle on the half-line in a harmonic plus inverse-square potential is shown to be a simple exercise. Second, the first rigorous derivation of the energy-dependent Green function of the one-dimensional Morse system is given.
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.