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Physical Review A

Test of a path-integral approach for the computation of scattering cross sections on an exactly solvable model

Valery V Smirnov

Published Nov 13, 2007• 8 citations
Publisher

Abstract

A numerical approach in scattering theory based on the path-integration technique is considered. An evaluation of a discretized Feynman path integral with an accurate Gaussian expansion of the integrand was used for the calculation of the differential cross sections for the scattering of a wave packet by a three-dimensional spherically symmetric Gaussian potential. The results are compared with the precise calculations using the phase functions method. This reveals separately discretization errors of the path integral, which depends on the single parameter---the number of discrete path points. High accuracy of the approach is demonstrated.

DOI
10.1103/physreva.76.052706

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A numerical approach in scattering theory based on the path-integration technique is considered. An evaluation of a discretized Feynman path integral with an accurate Gaussian expansion of the integrand was used for the calculation of the differential cross sections for the scattering of a wave packet by a three-dimensional spherically symmetric Gaussian potential. The results are compared with the precise calculations using the phase functions method. This reveals separately discretization errors of the path integral, which depends on the single parameter---the number of discrete path points. High accuracy of the approach is demonstrated.
Towards the Classification of Exactly Solvable Feynman Path Integrals: δδδ-Function Perturbations and Boundary-Problems as Miscellaneous Solvable Models
Invited talk given at the ``International Workshop on `Symmetry Methods in Physics' in memory of Ya.\ A.\ Smorodinsky, 5--10 July 1993, Dubna, Russia; to appear in the proceedings. In this contribution I present further results on steps towards a Table of Feynman Path Integrals. Whereas the usual path integral solutions of the harmonic oscillator (Gaussian path integrals), of the radial harmonic oscillator (Besselian path integrals), and the (modified) Pöschl-Teller potential(s) (Legendrian path integrals) are well known and can be performed explicitly by exploiting the convolution properties of the various types, a perturbative method opens other possibilities for calculating path integrals. Here I want to demonstrate the perturbation expansion method for point interactions and boundary problems in path integrals.
Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. More importantly, quantum algorithms for calculating path integrals require real-time evolution.Purpose: We test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals in order to understand the challenges facing real-time path integral calculations of scattering observables.Method: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on cylinder sets of continuous paths with respect to a complex probability distribution. The method can in principle be applied to arbitrary short-range potentials.Results: The method is applied to compute matrix elements of M\o{}ller wave operators applied to narrow wave packets. These are used to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half-shell transition operator matrix elements converge to the numerical solution of the Lippmann-Schwinger equation.Conclusions: This work presents a proof in principle that scattering observables can be computed using real-time Feynman path integrals. While the computational method is not efficient, it can be improved. It provides a laboratory for studying quantum computational algorithms that are applicable to scattering problems.
An expansion method in the path-integral formulation of quantum mechanics, as proposed in a previous paper, is extended to account for states with odd parity. The method is tested on the case of a one-dimensional attractive delta-function potential and the well-known solutions for both bound and scattering states are obtained analytically by an exact summation of the expansion series. The applicability of the method is discussed and it is shown how the energy spectrum could be determined by means of Stieltjes theory of moments.
In this master thesis, a new approximation scheme to non-relativistic potential scattering is developed and discussed. The starting points are two exact path integral representations of the T-matrix, which permit the application of the Feynman-Jensen variational method. A simple Ansatz for the trial action is made, and, in both cases, the variational procedure singles out a particular one-particle classical equation of motion, given in integral form. While the first is real, in the second representation this trajectory is complex and evolves according to an effective, time dependent potential. Using a cumulant expansion, the first correction to the variational approximation is also evaluated. The high energy behavior of the approximation is investigated, and is shown to contain exactly the leading and next-to-leading order of the eikonal expansion, and parts of higher terms. Our results are then numerically tested in two particular situations where others approximations turned out to be unsatisfactory. Substantial improvements are found.
The evaluation of a Feynman path integral is discussed for a scattering problem in which some degrees of freedom of the system are treated classically, while others are treated quantum mechanically. The path integral is of the type $K(0, {t}^{\ensuremath{'}\ensuremath{'}}; 0, {t}^{\ensuremath{'}})$, and is of interest in a multichannel semiclassical collision theory previously developed by Pechukas. It is shown that this path integral can be evaluated precisely in practice, despite the presence of noncausal terms in the integrand.
It is proposed that each of several analytical techniques used to evaluate the path integral when a spinless particle encounters an infinite potential barrier relies on more constraints than is necessary. We review the image, forbidden-paths, and eigenfunction-expansion methods, and the infinitesimal-propagator method is developed herein. It is then shown that the latter two methods contain assumptions leading to extra terms in the path-integral integrand which serve only to aid in the analytical evaluation of the path integral but which do not affect the outcome of that evaluation. The analytical arguments are further confirmed by a recently developed numerical method that, for a limited number of cases, is a direct evaluation of the path integral.