Stochastic methods for zero energy quantum scattering

Abstract

We investigate the use of stochastic methods for zero energy quantum scattering based on a path integral approach. With the application to the scattering of a projectile from a nuclear many body target in mind, we use the potential scattering of a particle as a test for the accuracy and efficiency of several methods. To be able to deal with complex potentials, we introduce a path sampling action and a modified scattering observable. The approaches considered are the random walk, where the points of a path are sequentially generated, and the Langevin algorithm, which updates an entire path. Several improvements are investigated. A cluster algorithm for dealing with scattering problems is finally proposed, which shows the best accuracy and stability.

Exact Path-Integral Representations for the T-Matrix in Nonrelativistic Potential Scattering

“…We start with the definition of the S-matrix as matrix element of the time-evolution operator in the interaction picture at infinite scattering times (1) and its connection with the T -matrix. Here the scattering states are normalized according to φ f |φ i = (2π) 3 δ (3) (k f − k i ) .…”

“…Energy conservation is not yet evident. 1 Better done in the discretized version of the path integral which shows that one has N v-integrations compared to N −1 ones over the intermediate points of the trajectory when the endpoints are fixed.…”

“…Also the long-standing problem how to evaluate real-time path integrals by stochastic methods needs a suitable path integral representation as starting method. There has been significant progress in dealing with real-time path integrals for dissipative systems but in closed systems and infinite scattering times only zero-energy scattering seems to be tractable by Euclidean Monte-Carlo methods at present [1].…”