We apply our recently proposed finite-difference Heisenberg (FDH) approach to the treatment of unbound states and show that, by using this approach, the problem of artificial reflections of the scattered wave packet from the boundaries is totally eliminated. This is because the basis and coordinate frame are dynamic and thus adjust themselves, as dictated by the potential function, to the evolving wave function. The disadvantage of this approach is that it may scale as M(3) for M basis and thus become expensive if M is large. However, accurate results can be obtained efficiently for a limited range of energies in the scattered wave packet using a small basis. These results are demonstrated by one-dimensional (1D) scattering from an Eckart barrier.