In a one-dimensional system the contact -potential is a ``good'' potential
and the problem of a scattering on it is solved in a standard manner, as described
in Sec. 1.3.3.1 without any special tricks. The situation is different in
three-dimensions where the
-potential has to be regularized (refer to
Sec.1.3.4.1) in order to avoid a possible divergence which can be caused by
the behavior of a symmetric solution (1.39).
The -pseudopotential turns out to be highly useful theoretical tool. Indeed
the commonly used Gross-Pitaevskii equation corresponds to pseudopotential
interaction
. A system of particles with
-pseudopotential interaction (5.1) is one of few exactly solvable one
dimensional quantum systems.
The Schrödinger equation (1.61) of the scattering on a pseudopotential
In the region it takes form of a free particle propagation
with the even solution given by
We are left with the only point , where the scattering potential is nonzero
. The infinite strength of the
-potential makes the first
derivative of the potential be discontinuous. Indeed, the proper boundary condition
can be obtained by integrating the equation (1.64) from infinitesimally
small
up to
. The integral of the continuous function
is proportional to
and vanishes in the limit
. Instead the
-function extracts the value of the function in zero and one
obtains the relation
This boundary condition for the solution (1.65) provides a relation between
the scattering phase and the momentum
of an incident particle
Taking the limit of the low energy scattering from (1.63) one obtains the
value of the scattering length
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(1.68) |
This expression can be read the other around: for equal mass particles
the strength of the potential
in a one-dimensional homogeneous system is
related to the value of the one-dimensional coupling constant as
It is interesting to note, that the sign in the relation of the scattering length to
the coupling constant is opposite to the one of a three dimensional system. In
positive scattering length corresponds to repulsion and negative one to attraction.
Another difference is that the 3D coupling constant is directly proportional to the
scattering length, although
is inversely proportional to
.
In terms of the phase (1.67) becomes
In the low energy limit the phase (1.70) can be expanded
and the scattering solution becomes
simply
. One sees that the one-dimensional scattering length
coincides with the position of the first node of the analytic continuation of the
low-energy solution1.7.