dorsal/arxiv
View SchemaRigged Hilbert Space Treatment of Continuous Spectrum
| Authors | R. de la Madrid, A. Bohm, M. Gadella |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0109154 |
| URL | https://arxiv.org/abs/quant-ph/0109154 |
| DOI | 10.1002/1521-3978(200203)50:2<185::AID-PROP185>3.0.CO;2-S |
| Journal | Fortschr. Phys. 50 (2002) 185-216 |
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 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.
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"abstract": "The ability of the Rigged Hilbert Space formalism to deal with continuous\nspectrum is demonstrated within the example of the square barrier potential.\nThe non-square integrable solutions of the time-independent Schrodinger\nequation are used to define Dirac kets, which are (generalized) eigenvectors of\nthe Hamiltonian. These Dirac kets are antilinear functionals over the space of\nphysical wave functions. They are also basis vectors that expand any physical\nwave function in a Dirac basis vector expansion. It is shown that an acceptable\nphysical wave function must fulfill stronger conditions than just square\nintegrability--the space of physical wave functions is not the whole Hilbert\nspace but rather a dense subspace of the Hilbert space. We construct the\nposition and energy representations of the Rigged Hilbert Space generated by\nthe square barrier potential Hamiltonian. We shall also construct the unitary\noperator that transforms from the position into the energy representation. We\nshall see that in the energy representation the Dirac kets act as the\nantilinear Schwartz delta functional. In constructing the Rigged Hilbert Space\nof the square barrier potential, we will find a systematic procedure to\nconstruct the Rigged Hilbert Space of a large class of spherically symmetric\npotentials. The example of the square barrier potential will also make apparent\nthat the natural framework for the solutions of a Schrodinger operator with\ncontinuous spectrum is the Rigged Hilbert Space rather than just the Hilbert\nspace.",
"arxiv_id": "quant-ph/0109154",
"authors": [
"R. de la Madrid",
"A. Bohm",
"M. Gadella"
],
"categories": [
"quant-ph",
"hep-th",
"math-ph",
"math.MP"
],
"doi": "10.1002/1521-3978(200203)50:2\u003c185::AID-PROP185\u003e3.0.CO;2-S",
"journal_ref": "Fortschr. Phys. 50 (2002) 185-216",
"title": "Rigged Hilbert Space Treatment of Continuous Spectrum",
"url": "https://arxiv.org/abs/quant-ph/0109154"
},
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