  We use the matrix product state formalism to construct stationary scattering
states of elementary excitations in generic one-dimensional quantum lattice
systems. Our method is applied to the spin-1 Heisenberg antiferromagnet, for
which we calculate the full magnon-magnon S matrix for arbitrary momenta and
spin, the two-particle contribution to the spectral function and the
magnetization curve. As our method provides an accurate microscopic
representation of the interaction between elementary excitations, we envisage
the description of low-energy dynamics of one-dimensional spin chains in terms
of these particlelike excitations.
