It is known that Schrödinger equation for the wavefunction of two colliding particles can be decomposed in one equation that describes the center-of-mass motion, a free particle equation with mass
, and another that describes the relative motion. With the definition
for the reduced mass, in the relative frame the Schödinger equation reads
Let us suppose that the potential is short-ranged, so that there exists a characteristic length
such that the potential can be neglected outside this radius,
In this case Equation (
12) becomes a free Schrödinger equation whose solution is given by the composition of an incoming plane wave with momentum
and a scattering state with momentum
. It is relevant to look for elastic collisions, that can be studied fixing the scattering energy to
, equal to the incoming wave energy. The solution is given by
The homogeneous solution (
) is simply
, therefore, it is possible to write an implicit solution, known as the Lippmann–Schwinger equation
with
an infinitesimally small positive real number. In the coordinate representation, it reads
At long distances, (see
Appendix A for details) Equation (
16) can be written as
after defining
, and the scattering amplitude
The scattering solution at large distances is then composed by an incoming plane wave and an outgoing spherical wave, weighted by the scattering amplitude. In terms of the latter it is also possible to define the differential cross section of the process, at large distances,
Due to the spherically symmetric potential, the scattering amplitude depends only on the modulus
p and the angle
. It is convenient, therefore, to perform a spherical wave expansion
with
the Legendre polynomials. In order to preserve the normalization of the wavefunction, one has to impose the following condition for the coefficients
(see
Appendix B)
which defines the phase shift
and where we rescaled, here and in what follows,
for simplicity, to get rid of
ℏ. This relation imposes that when the relative momentum vanishes, also the phase shift should be zero,
. The scattering amplitude, at fixed angular momentum, can be rewritten as
In the radial Schrödinger equation (see
Appendix C), for
, besides the potential
there is also the so-called centrifugal barrier, that depends on the angular momentum at which the scattering happens and on the relative distance. In ultracold atoms, at sufficiently low temperatures, the atoms may not have enough energy to overcome this centrifugal barrier, therefore the only relevant contribution to the scattering amplitude will come from s-wave scattering (
). To clarify this point one can estimate the angular momentum as the product of the range of the interaction
times the momentum given by the inverse of the thermal de Broglie length
. For ultracold gases the temperatures are typically
and, since the thermal wavelength
, it is allowed to consider
, namely, the scattering amplitude is dominated by the s-wave contribution
It is expected that scattering processes modified by the presence of
for
are greatly suppressed at sufficiently low energy. Moreover, also the
contribution is affected by the low energy hypothesis. As shown in Ref. [
76], it is possible to define the scattering length at fixed
ℓ considering the low energy limit
which comes from the low energy limit of the phase shift
This can be derived by solving the Schrödinger equation as shown in
Appendix C. It is remarkable that ultracold gases can be cooled down to the point where only one partial wave in the two-body problem becomes dominant, the s-wave contribution. At low energies (
) it is possible to expand the phase
, so defining the scattering length as
Actually only
has the dimension of a length, contrary to the other terms
with
. Considering the relation in Equation (
22) and the following expansion
the scattering amplitude is then given, at low energy, by the expression
which shows explicitly that for ultracold gases, under the hypothesis of short-range central potential, at low energy, the two-body scattering depends only on two parameters:
, the s-wave scattering length, and
, an effective range proportional to
. We will not treat
, the effective range of the potential, more deeply here, because, for what follows, only the first order expansion of the phase shift is needed. Finally also the cross section admits a low energy limit. Writing
, we have
and
. In particular, from Equations (
A15) and (
A17), one gets
for spinless bosons,
for spinless (polarized) fermions,
for fermions in a singlet state,
for fermions in a triplet state. These results show that the interaction between polarized fermions is suppressed, while interaction in the singlet channel is dominant in the low energy limit, consistently with the Pauli exclusion principle.