  We construct a discrete non-hermitean momentum operator, which implements
faithfully the non self-adjoint nature of momentum for a particle in a box. Its
eigenfunctions are strictly limited to the interior of the box in the continuum
limit, with the quarter wave as first non-trivial eigenstate. We show how to
construct the corresponding hermitean Hamiltonian for the infinite well as
concrete example to realize unitary dynamics. The resulting Hilbert space can
be decomposed into a physical and unphysical subspace, which are mutually
orthogonal. The physical subspace in the continuum limit reproduces that of the
continuum theory and we give numerical evidence that the correct probability
distributions for momentum and energy are recovered.
