Approximate Methods in Many-body Scattering

Theoretical Physics

Julien Dujardin, Thomas Engl, Juan Diego Urbina +1 more

We consider quasi‐stationary scattering of interacting bosonic matter waves in one‐dimensional waveguides, as they arise in guided atom lasers. We show how the truncated Wigner (tW) method, which corresponds to the semiclassical description of the bosonic many‐body system on the level of the diagonal approximation, can be utilized in order to describe such many‐body bosonic scattering processes. Special emphasis is put on the discretization of space at the exact quantum level, in order to properly implement the semiclassical approximation and the tW method, as well as on the discussion of the results to be obtained in the continuous limit. image

T. T. Chou, Chen Ning Yang, L. H. Yü

We point out that the local density approximation (LDA) of Oliva is an adaptation of the Thomas-Fermi method, and is a good approximation when \ensuremath{\epsilon}=\ensuremath{\Elzxh}\ensuremath{\omega}/kT\ensuremath{\ll}1. For the case of scattering length a\ensuremath{\gtrsim}0, the LDA leads to a quantitative result, Eq. (14\ensuremath{'}), easily checked by experiments. Critical remarks are made about the physics of the many-body problem in terms of the scattering length a. \textcopyright{} 1996 The American Physical Society.

Y. Alhassid, S. E. Koonin

A nonperturbative method is developed for calculating the excitation of a many-body system by a time-dependent Hamiltonian. The stationary-phase approximation to a functional-integral representation of the interaction-picture many-body evolution operator results in a mean-field approximation to the $S$ matrix which is asymptotically time independent. A one-body temporally nonlocal evolution equation defines the stationary mean-field configurations. The general method and character of the stationary solutions are illustrated by application to the forced harmonic oscillator and forced Lipkin model. Potential applications to realistic nuclear and atomic scattering situations are discussed.NUCLEAR REACTIONS Mean-field approximation, many-body $S$ matrix, time-dependent Hartree Fock.

Israel Michael Sigal, Avy Soffer