  The Coordinate Bethe Ansatz (CBA) expresses, as a sum over permutations, the
matrix element of an XXX Heisenberg spin chain Hamiltonian eigenstate with a
state with fixed spins. These matrix elements comprise the wave functions of
the Hamiltonian eigenstates. However, as the complexity of the sum grows
rapidly with the length N of the spin chain, the exact wave function in the
continuum limit is too cumbersome to be exploited. In this note we provide an
approximation to the CBA whose complexity does not directly depend upon N. This
consists of two steps. First, we add an anchor to the argument of the
exponential in the CBA. The anchor is a permutation-dependent integral multiple
of 2 pi. Once anchored, the distribution of these arguments simplifies,
becoming approximately Gaussian. The wave function is given by the Fourier
transform of this distribution and so the calculation of the wave function
reduces to the calculation of the moments of the distribution. Second, we
parametrize the permutation group as a map between integers and we bin these
maps. The calculation of the moments then reduces to a combinatorial exercise
on the partitioning into bins. As an example we consider the matrix element
between the classical and quantum ground states.
