Harmonic oscillator with nonzero minimal uncertainties in both position and momentum in a SUSYQM framework
Abstract
In the context of a two-parameter (agr, bgr) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters agr, bgr vanishes, in which case shape invariance under parameter translation occurs. In the special case where agr = bgr \not= 0, the oscillator Hamiltonian is shown to coincide with that of the q-deformed oscillator with q > 1 and its eigenvectors are therefore n-q-boson states. In the general case where 0 \not= agr \not= bgr \not= 0, the eigenvectors are constructed as linear combinations of n-q-boson states by resorting to a Bargmann representation of the latter and to q-differential calculus. They are finally expressed in terms of a q-exponential and little q-Jacobi polynomials.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- October 2003
- DOI:
- 10.1088/0305-4470/36/41/009
- arXiv:
- arXiv:math-ph/0306047
- Bibcode:
- 2003JPhA...3610373Q
- Keywords:
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- Harmonic oscillatorUncertainty relationsq-DeformationsBasic special functions;
- Mathematical Physics;
- High Energy Physics - Theory;
- Mathematics - Mathematical Physics;
- Mathematics - Quantum Algebra;
- Quantum Physics
- E-Print:
- LaTeX, 24 pages, no figure, minor changes, additional references, final version to be published in JPA