In order to test the universality assumption and if the details of the potential are
important it is useful to have a potential, where the range of the potential can
be varied while keeping the -wave scattering length constant. In the case of the
hard-sphere (Sec. 1.3.2.2) both distances are the same. The easiest way to
modify the hard sphere potential (1.48) in such a way that it has desired
properties is to make the height of the potential finite. The resulting potential is
called the soft-sphere (SS) potential:
The Schrödinger equation (1.41) for a pair of particles in the center of
mass system is given by
We are interested in scattering at small energy, so . For convenience we
introduce
, where is real. The second equation out of the
pair (1.53) has a free wave solution which extends with the same amplitude to
large distances, although the first equation has a decaying solution expressed in
terms of the hyperbolic sinus:
The phase must be equal to zero in order to obtain a solution which is
not divergent at (see condition 1.40). We impose continuity of solution
and its derivative in the point :
Condition of the continuity of the logarithmic derivative
fixes the phase of the solution:
(1.58) |
By taking limit of low energy in (1.57) and using the definition
(1.44) one obtains the expression for the -wave scattering length
for the scattering on the SS potential:
If in the case of the hard core potential (1.48) the potential energy is absent, it is no longer so here. This makes it reasonable to use a pair of potentials SS-HS in order to test the universality of the -wave description (see, e.g., study done in Chapter. 3).