Rigged Hilbert Space Treatment of Continuous Spectrum
Abstract
The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The nonsquare integrable solutions of the timeindependent 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 integrabilitythe 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.
 Publication:

Fortschritte der Physik
 Pub Date:
 March 2002
 DOI:
 10.1002/15213978(200203)50:2<185::AIDPROP185>3.0.CO;2S
 arXiv:
 arXiv:quantph/0109154
 Bibcode:
 2002ForPh..50..185D
 Keywords:

 Quantum Physics;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 RevTex, 32 pages