Let us familiarize ourselves with these scattering amplitudes by
examining low energy scattering in the
In this scenario we have only one partial wave contributing
(obviously) with a phase shift .
The cross section for S-wave scattering is then given in terms
of this one phase shift
In the limit of
as we have been considering above,
the solution to the schrodinger equation
in the region where the potential vanishes
One of the things we should learn from this is that all we see in the low energy (compared to the inverse radius of the potential) is the asymptotic behaviour of the wavefunction characterized by one constant, the scattering length. This is a nontrivial construction from the potential itself and the wavefunction of the state. We see that for momenta much less than the inverse radius of the potential the scattering length is sufficient to describe all of the interactions. This goes back to our original ideas about short distance physics being encoded in the constants of the low energy theory. It is clear that by measuring the scattering length of a system alone we cannot reconstruct the potential uniquely. There are infinitely many different shapes, depths and ranges of potentials that will reproduce a single scattering length.
Let us consider the system I have shown in the
It is important to notice that for the wavefunction inside this potential must have just turned over to have a negative derivative at the boundary. This would indicate that a bound state exists for this potential. In order for us to arrive at this conclusion we see that the existence of a bound state requires that the wavefunction at the boundary have a negative derivative so that we can match onto an exponentially falling wavefunction outside the potential. Such a fall off is characteristic of a state with negative energy , and is localized about the centre of the potential. We will return to this later on. Also, we note that if the scattering length were negative , then the wavefunction would have a positive derivative at the boundary of the potential. This indicates that there is not a loosely bound () state for this system.
So far we have considered
scattering and found that a single
parameter can be identified, the
scattering length, and we have discussed what this means in terms of
wavefunctions and cross sections.
However, when we move away from
scattering and ask about the
cross section for non-zero momenta
but still much smaller than the inverse radius, then we expect to be
able to perform a taylor expansion on
the scattering amplitude, or some quantity related to the
We have observed in our previous discussions that we can "encode"
(in some complicated way) the potential and
wavefunction in this region by the phase shift of the eigenstates
of the potential. We expect that the
dependence on small momenta (compared with the potential range)
can also be encoded by this phase shift.
We therefore wish to form a derivative expansion of the phase
shift, or some function of the phase shift, if
it is more appropriate.
be the asymptotic form of
where the radial wavefunction is
We have already seen that
and we can form the quantity
Before we proceed, lets look at this object a slightly different way.
Starting with the scattering amplitude
for an S-wave interaction
An interesting feature of , is that when the entire denominator of vanishes, the scattering amplitude becomes infinite, corresponding to the existance of a bound state at that location. Therefore, solving determines the location in space of bound states in either channel. In the channel, this corresponds to the deuteron bound state with a binding energy of . In the channel the pole has positive energy corresponding to a resonance in the scattering amplitude. In fact, the location of the deuteron pole is one of the inputs into the precise determination of and .