  The usual derivations of the S and K matrices for two-particle reactions
proceed through the Lippmann-Schwinger equation with formal definitions of the
incoming and outgoing scattering states. Here we present an alternative
derivation that is carried out completely in the Hamiltonian representation,
using a discrete basis of configurations for the scattering channels as well as
the quasi-bound configurations of the combined fragments. We use matrix algebra
to derive an explicit expression for the K matrix in terms of the Hamiltonian
of the internal states of the compound system and the coupling between the
channels and the internal states. The formula for the K matrix includes
explicitly a real dispersive shift matrix to the internal Hamiltonian that is
easily computed in the formalism. That expression is applied to derive the
usual form of the S matrix as a sum over poles in the complex energy plane.
Some extensions and limitations of the discrete-basis Hamiltonian formalism are
discussed in the concluding remarks and in the Appendix.
