In this section we will formulate the generic scattering problem in a three-dimensional space. At low density the interaction between particles of a gas is well described by binary collisions.
Consider a collision of two particles having coordinates and , masses
and . The aim is to find the stationary solution
of the
Schrödinger equation:
In absence of an external confinement the problem is translationary
invariant and the center of mass moves with a constant velocity. The solution of
problem gets separated in the center of the mass frame. The Schrödinger equation
for the movement of the center of mass
is trivial:
The equation for the relative coordinate
involves the interaction
potential:
In order to proceed further we will assume that the energy of the incident particle
is small and the solution has a spherical symmetry
.
In this case the Laplacian gets simplified
and the Eq. 1.36 is
conveniently rewritten by introducing function
The solution of this equation in a general form can be written as
The scattering at low energy (which describes well a binary collisions in a dilute
gas) has a special interest as it becomes universal and can be described in terms of
one parameter, the -wave scattering length :
In the asymptotic limit of slow particles the scattering solution
(1.42) can be expanded
In the next several sections we will solve the problem of the scattering on a hard-sphere potential (1.3.2.2) and a soft-sphere potential (1.3.2.3). We will find explicit expressions for the scattered functions, which are of a great importance, as in many cases can provide a physical insight into properties of a many body problem. Indeed, at a certain conditions the correlation functions can be related to the scattered function . Another point is that the two-body Bijl-Jastrow term (2.37) in the construction of the trial wave function is very often taken in a form of . Thus such calculations are very important for the implementation of the Quantum Monte Carlo methods.
We will also find expressions for the scattering length in terms of the
height (or depth) of the potential :