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 non-square integrable solutions of the time-independent Schrödinger 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 Schrödinger operator with continuous spectrum is the Rigged Hilbert Space rather than just the Hilbert space.

Reply to ‘Comment on ‘On the inconsistency of the Bohm–Gadella theory with quantum mechanics’’

“…The theory of distributions [4] says that a test function ϕ(r) on which a distribution d(r) acts is such that the following integral is finite: ‡ 13) where ϕ|d represents the action of the functional |d on the test function ϕ. With some variations, this is the "standard method" followed by [7][8][9][10][11][12][13][14][15] to introduce spaces of test functions in quantum mechanics. Thus, contrary to what the authors of [1] assert, the method followed by the present author runs (somewhat) parallel to [8], not to TAQT.…”

The Friedrichs model and its use in resonance phenomena

“…In our case, the vector space of test functions should have certain analyticity conditions so that the antiduals contain the Gamow vectors. Several motivations (see [24,4,23,29,25,27,26,41]) suggest the use of functions which are at the same time Hardy and Schwartz, so that the spaces of test vectors are given by 22) where S denotes the Schwartz space, H 2 ± the space of Hardy functions on the upper (with +) and lower (with −) half planes. These functions are restricted to the positive semiaxis because they represent wave functions in the energy representation and we assume that the energy is always positive.…”

The role of the rigged Hilbert space in quantum mechanics

“…[19,20,21], has been applied to two simple three-dimensional potentials, see Refs. [27,28], to the three-dimensional free Hamiltonian, see Ref. [29], and to the 1D rectangular barrier potential, see Ref.…”