  The ability of the Rigged Hilbert Space formalism to deal with continuous
spectrum is demonstrated within the example of the square barrier potential.
The non-square integrable solutions of the time-independent Schrodinger
equation are used to define Dirac kets, which are (generalized) eigenvectors of
the Hamiltonian. These Dirac kets are antilinear functionals over the space of
physical wave functions. They are also basis vectors that expand any physical
wave function in a Dirac basis vector expansion. It is shown that an acceptable
physical wave function must fulfill stronger conditions than just square
integrability--the space of physical wave functions is not the whole Hilbert
space but rather a dense subspace of the Hilbert space. We construct the
position and energy representations of the Rigged Hilbert Space generated by
the square barrier potential Hamiltonian. We shall also construct the unitary
operator that transforms from the position into the energy representation. We
shall see that in the energy representation the Dirac kets act as the
antilinear Schwartz delta functional. In constructing the Rigged Hilbert Space
of the square barrier potential, we will find a systematic procedure to
construct the Rigged Hilbert Space of a large class of spherically symmetric
potentials. The example of the square barrier potential will also make apparent
that the natural framework for the solutions of a Schrodinger operator with
continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert
space.
