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arXiv:1411.3263 · Journal of Physics Conference Series

Quantum physics and signal processing in rigged Hilbert spaces by means of special functions, Lie algebras and Fourier and Fourier-like transforms

E Celeghini, M. A. del Olmo

Published Apr 13, 2015• 5 citations
arXivHTMLPDF

Abstract

Quantum physics and signal processing in the line R are strictly related to Fourier transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io (2). A rigged Hilbert space is found and a new discrete basis in R obtained. All operators defined on R are shown to belong to the universal enveloping algebra of io (2) allowing, in this way, their algebraic treatment. Introducing in the half-line a Fourier-like transform, the procedure is extended to R + and can be easily generalized to R n and to spherical cohordinate systems.

DOI
10.1088/1742-6596/597/1/012022

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Quantum physics and signal processing in the line R are strictly related to Fourier transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io (2). A rigged Hilbert space is found and a new discrete basis in R obtained. All operators defined on R are shown to belong to the universal enveloping algebra of io (2) allowing, in this way, their algebraic treatment. Introducing in the half-line a Fourier-like transform, the procedure is extended to R + and can be easily generalized to R n and to spherical cohordinate systems.
Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A Rigged Hilbert space is found and a new discrete basis in R obtained. The operators {O[R]} defined on R are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in this way, their algebraic discussion. Introducing in the half-line a Fourier-like Transform, the procedure is extended to R^+ and can be easily generalized to R^n and to spherical reference systems.
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 function 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.
We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous representations in a nested scale of Hilbert spaces. We also construct a couple of examples illustrative of the key features of group representations in rigged Hilbert spaces. Finally, we establish a simple criterion for the integrability of an operator Lie algebra in a rigged Hilbert space.
Quantum mechanics of a linear harmonic oscillator in a finite-dimensional Hilbert space satisfying the correct equations of motion is studied. The connections to Weyl's formulation of the algebra of bounded unitary operators in finite space as well as to a truncated quantized linear harmonic oscillator are discussed. It is pointed out that the discrete Fourier transformation (DFT) plays a central role in determining the actual form of the position, the momentum, the number and the Hamiltonian operators. The explicit form of these operators in different bases is exhibited for some low values of the dimension of the Hilbert space. In this formulation, it is shown that the Hamiltonian is indeed the logarithm of the DFT and that by modifying Weyl's framework to include position and momentum operators with non-uniformly spaced spectra the equations of motion are satisfied.
Roberts' proposal of a rigged Hilbert space Φ⊂G⊂Φ× for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces which might be of interest for the application of this formalism to special quantum systems. It is shown that on the basis of this proposal one also obtains a satisfactory solution for a rigged Hilbert space for composite systems. Another part is concerned with topological properties of the so-called eigenoperators γ(t) belonging to an A-eigen-integral decomposition of Φ with respect to a self-adjoint operator A on G. We derive a representation of γ(t) in terms of the generalized eigenvectors of A and in the same context give a rough topological characterization of these eigenvectors.
It is shown that the natural framework for the solutions of any Schrödinger equation whose spectrum has a continuous part is the Rigged Hilbert Space rather than just the Hilbert space. The difficulties of using only the Hilbert space to handle unbounded Schrödinger Hamiltonians whose spectrum has a continuous part are disclosed. Those difficulties are overcome by using an appropriate Rigged Hilbert Space (RHS). The RHS is able to associate an eigenket to each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum. The collection of eigenkets corresponding to both discrete and continuous spectra forms a basis system that can be used to expand any physical wave function. Thus the RHS treats discrete energies (discrete spectrum) and scattering energies (continuous spectrum) on the same footing. Typeset using REVTEX 1