dorsal/arxiv
View SchemaApplication of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0506094 |
| URL | https://arxiv.org/abs/quant-ph/0506094 |
| DOI | 10.1063/1.2063168 |
| Journal | J. Math. Phys. 46, 102108 (2005) |
Abstract
We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modelling the propagation of electromagnetic waves travelling in a wave guide half and half filed with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter zeta, we show that the equivalent Hermitian Hamiltonian has the form $\frac{p^2}{2m}+\frac{\zeta^2}{2}\sum_{n=0}^\infty\{\alpha_n(x),p^{2n}\}$ with $\alpha_n(x)$ vanishing outside an interval that is three times larger than the support of $v(x)$, i.e., in 2/3 of the physical interaction region the potential $v(x)$ vanishes identically. We provide a physical interpretation for this unusual behavior and comment on the classical limit of the system.
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"abstract": "We extend the application of the techniques developed within the framework of\nthe pseudo-Hermitian quantum mechanics to study a unitary quantum system\ndescribed by an imaginary PT-symmetric potential v(x) having a continuous real\nspectrum. For this potential that has recently been used, in the context of\noptical potentials, for modelling the propagation of electromagnetic waves\ntravelling in a wave guide half and half filed with gain and absorbing media,\nwe give a perturbative construction of the physical Hilbert space, observables,\nlocalized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of\norder three or higher in the non-Hermiticity parameter zeta, we show that the\nequivalent Hermitian Hamiltonian has the form\n$\\frac{p^2}{2m}+\\frac{\\zeta^2}{2}\\sum_{n=0}^\\infty\\{\\alpha_n(x),p^{2n}\\}$ with\n$\\alpha_n(x)$ vanishing outside an interval that is three times larger than the\nsupport of $v(x)$, i.e., in 2/3 of the physical interaction region the\npotential $v(x)$ vanishes identically. We provide a physical interpretation for\nthis unusual behavior and comment on the classical limit of the system.",
"arxiv_id": "quant-ph/0506094",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.2063168",
"journal_ref": "J. Math. Phys. 46, 102108 (2005)",
"title": "Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States",
"url": "https://arxiv.org/abs/quant-ph/0506094"
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