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

Joachim Seifert

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 403-413
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Seifert, Joachim. "Generalized hermite polynomials obtained by embeddings of the q-Heisenberg algebra." Banach Center Publications 40.1 (1997): 403-413. <http://eudml.org/doc/252243>.

@article{Seifert1997,
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.},
author = {Seifert, Joachim},
journal = {Banach Center Publications},
keywords = {-phase-space; -oscillator realization; -Heisenberg algebra; Stieltjes Wigert polynomials; Schrödinger equation; difference equations},
language = {eng},
number = {1},
pages = {403-413},
title = {Generalized hermite polynomials obtained by embeddings of the q-Heisenberg algebra},
url = {http://eudml.org/doc/252243},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Seifert, Joachim
TI - Generalized hermite polynomials obtained by embeddings of the q-Heisenberg algebra
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 403
EP - 413
AB - 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.
LA - eng
KW - -phase-space; -oscillator realization; -Heisenberg algebra; Stieltjes Wigert polynomials; Schrödinger equation; difference equations
UR - http://eudml.org/doc/252243
ER -

References

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  1. [1] J. Schwenk, J. Wess, A Quantum Mechanical toy model, Phys. Lett., B 291 (1992) 273. 
  2. [2] A. Hebecker, S. Schreckenberg, J. Schwenk, W. Weich, J. Wess, Representations of a q-deformed Heisenberg Algebra, MPI-Ph/93-45, (1993). 
  3. [3] M. Fichtmueller, A. Lorek and J. Wess, Q-deformed Phase Space and its Lattice Structure, MPI-PhT/95-109. 
  4. [4] Tom H. Koornwinder, Orthogonal Polynomials in Connection with Quantum Groups, P. Nevai (ed.), Orthogonal Polynomials,Kluwer Academic Publishers, (1990) 257-292. Zbl0697.42019
  5. [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. [6] Gasper and Rahman, Basic Hypergeometric Series, Cambridge University Press, (1990). Zbl0695.33001
  7. [7] A. Macfarlane, On q-analogues of the quantum harmonic oscillator and the quantum group S U q ( 2 ) , J. Phys., A 22 (1989) 4581. Zbl0722.17009
  8. [8] P. P. Kulish, On Recent Progress in Quantum Groups an introductory review, Jahrbuch Überblicke Mathematik 1993, Vieweg, (1993) 97. Zbl0778.17008
  9. [9] T. Curtwright, C. Zachos, Paradigms of Quantum Algebras, ANL-HEP-PR-90-61, (1990). 
  10. [10] Gaetano Fiore, The S O q ( N , ℝ ) -Symmetric Harmonic Oscillator on the Quantum Euclidean Space ℝ q N and It’s Hilbert Space Structure, International Journal of Modern Physics, Vol. 8, 26 (1993) 4679-4729. Zbl0985.81545
  11. [11] Joachim Seifert, Quantum Mechanical Representations of the Q-Oscillator, forthcoming publication. 

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