  The normalization of scattering states is more than a rote step necessary to
calculate expectation values. This normalization actually contains important
information regarding the density of the scattering spectrum (along with useful
details on the bound states). For many applications, this information is more
useful than the wavefunctions themselves. In this paper we show that this
correspondence between scattering state normalization and the density of states
is a consequence of the completeness relation, and we present formulas for
calculating the density of states which are applicable to certain potentials.
We then apply these formulas to the delta function potential and the square
well. We then illustrate how the density of states can be used to calculate the
partition function for a system of two particles with a point-like (delta
potential) interaction.
