Download PDF - Generalized hermite polynomials obtained by embeddings of the q-Heisenberg algebraAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Generalized hermite polynomials obtained by embeddings of the q-Heisenberg algebra

Authors 1

Affiliations

  1. Sektion Physik, Universität München, LS Prof. Wess, Theresienstr. 37, D-80333 München, Germany

Abstract

Several ways to embed q-deformed versions of the Heisenberg algebra into the classical algebra itself are presented. By combination of those embeddings it becomes possible to transform between q-phase-space and q-oscillator realizations of the q-Heisenberg algebra. Using these embeddings the corresponding Schrödinger equation can be expressed by various difference equations. The solutions for two physically relevant cases are found and expressed as Stieltjes Wigert polynomials.

Bibliography

  1. J. Schwenk, J. Wess, A Quantum Mechanical toy model, Phys. Lett., B 291 (1992) 273.
  2. A. Hebecker, S. Schreckenberg, J. Schwenk, W. Weich, J. Wess, Representations of a q-deformed Heisenberg Algebra, MPI-Ph/93-45, (1993).
  3. M. Fichtmueller, A. Lorek and J. Wess, Q-deformed Phase Space and its Lattice Structure, MPI-PhT/95-109.
  4. Tom H. Koornwinder, Orthogonal Polynomials in Connection with Quantum Groups, P. Nevai (ed.), Orthogonal Polynomials,Kluwer Academic Publishers, (1990) 257-292.
  5. Roelof Koekoek and René F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Reports of the faculty of technical Mathematics and Informatics no 94-05, (1994).
  6. Gasper and Rahman, Basic Hypergeometric Series, Cambridge University Press, (1990).
  7. A. Macfarlane, On q-analogues of the quantum harmonic oscillator and the quantum group SUq(2), J. Phys., A 22 (1989) 4581.
  8. P. P. Kulish, On Recent Progress in Quantum Groups an introductory review, Jahrbuch Überblicke Mathematik 1993, Vieweg, (1993) 97.
  9. T. Curtwright, C. Zachos, Paradigms of Quantum Algebras, ANL-HEP-PR-90-61, (1990).
  10. Gaetano Fiore, The SOq(N,ℝ)-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ℝN_q and It's Hilbert Space Structure, International Journal of Modern Physics, Vol. 8, 26 (1993) 4679-4729.
  11. Joachim Seifert, Quantum Mechanical Representations of the Q-Oscillator, forthcoming publication.
Banach Center Publications
1997/40/1
Export to PBN, Opens in new tab
Pages:
403-413
Main language of publication
English
Published
1997
Exact and natural sciences