Variational approximations in a path integral description of potential scattering

Abstract

Abstract. Using a recent path integral representation for the T -matrix in nonrelativistic potential scattering we investigate new variational approximations in this framework. By means of the Feynman-Jensen variational principle and the most general ansatz quadratic in the velocity variables -over which one has to integrate functionally -we obtain variational equations which contain classical elements (trajectories) as well as quantum-mechanical ones (wave spreading). We analyse these equations and solve them numerically by iteration, a procedure best suited at high energy. The first correction to the variational result arising from a cumulant expansion is also evaluated. Comparison is made with exact partial-wave results for scattering from a Gaussian potential and better agreement is found at large scattering angles where the standard eikonal-type approximations fail.

A new path-integral representation of the T-matrix in potential scattering

“…Our main aim in this work is to transcribe their method to the nonrelativistic case and to use the result for an approximate variational calculation similar to the one investigated in Refs. [9,10]. As a by-product a new path-integral representation of the scattering length is obtained.…”

“…For example, following Refs. [9,10] we may variationally approximate the velocity path integral in Eq. (28)…”

“…in the condensed notation of Ref. [10] where summation over discrete and continuous indices is implied. Evaluation of averages and derivation of the variational equations for the functions A ij (t, t ′ ), B i (t) follow the same lines as in Refs.…”

Стационарное Уравнение Шредингера Нерелятивистской Квантовой Механики И Функциональный Интеграл

“…Имеется ряд исследований (см., например, [13]- [16] и последнюю работу [17], где приведены ссылки на предыдущие статьи), в которых метод функционального ин-тегрирования применялся к теории потенциального рассеяния в нерелятивистской квантовой механике с целью получения представления для амплитуды рассеяния в форме интеграла по путям. Основная схема рассуждений состояла в том, чтобы ис-ходя из временного уравнения Шредингера написать представление для -матрицы как амплитуды перехода от состояний при = −∞ к состояниям при = ∞, а за-тем выделить -матрицу, из которой, в свою очередь, надо выделить -функцию, отвечающую за сохранение энергии в начальном и конечном состояниях.…”

Scattering theory with path integrals

“…[1] there has been renewed interest in the last few years to use it also for scattering problems [2,3,4] in nonrelativistic physics. This offers the chance of finding new approximation methods [5,6] or trying to evaluate the involved path integrals by stochastic methods although the oscillating nature of the real-time path integral presents a great challenge (see e.g. Ref.…”