2003/06/30 by C. Quesne, V. M. Tkachuk
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Non-Hermitian Physics #hep-th #math-ph #math.MP #math.QA #quant-ph
paper · pdf · doi:10.1088/0305-4470/36/41/009
published as J.Phys.A36:10373-10391,2003 · LaTeX, 24 pages, no figure, minor changes, additional references, final version to be published in JPA
arxiv created 2003/08/22 · openalex publication_date 2003/10/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
In the context of a two-parameter (α, β) 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 α, β vanishes, in which case shape invariance under parameter translation occurs. In the special case where α = β ≠ 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 ≠ α ≠ β ≠ 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.