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

Abstract

We employ the method used by Barbashov and collaborators in Quantum Field Theory to derive a pathintegral representation of the T -matrix in nonrelativistic potential scattering which is free of functional integration over fictitious variables as was necessary before. The resulting expression serves as a starting point for a variational approximation applied to high-energy scattering from a Gaussian potential. Good agreement with exact partial-wave calculations is found even at large scattering angles. A novel path-integral representation of the scattering length is obtained in the low-energy limit.

Scattering theory with path integrals

“…to parameters of the potential or the energy. This gave the same result as previously [3] derived from the field-theoretic work of Barbashov et al While this result is written in terms of velocity path integrals I also have succeeded to give a new representation by ordinary Feynman path integrals.…”

“…This is obvious since Eqs. (12), (8) or (24), (28) come all as a "level 1" -representation [3] of the scattering amplitude where one power of the potential appears in front of the path integral whereas an impact-parameter representation would belong to the "level 0" -class.…”

“…whereρ 3 The nomenclature indicates that ρ is a function of t, but a functional of v. 4 Of course, this integration could be performed analytically on the r.h.s. but for many applications the present form is preferable.…”

“…Following the pioneering work of Ref. [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.…”

“…an expression for the T -matrix was only possible by applying the Faddeev-Popov trick and introducing "anti-velocity" degrees of freedom which take away unphysical contributions in the path integral. In the meantime path-integral formulations for the T -matrix have been found (or rediscovered [3]) which do not need an "anti-velocity" at all but are more involved. They are either based on a technique invented by Barbashov and collaborators in quantum field theory [11] or use the asymptotic limit of the path-integral solution of the Schrödinger equation [4].…”