Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 093, 9 pages      arXiv:1212.0078
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Coherent States for Tremblay-Turbiner-Winternitz Potential

Yusuf Sucu and Nuri Unal
Department of Physics, Faculty of Science, Akdeniz University, 07058 Antalya, Turkey

Received July 31, 2012, in final form November 28, 2012; Published online December 01, 2012

In this study, we construct the coherent states for a particle in the Tremblay-Turbiner-Winternitz potential by finding the conserved charge coherent states of the four harmonic oscillators in the polar coordinates. We also derive the energy eigenstates of the potential and show that the center of the coherent states follow the classical orbits of the particle.

Key words: Tremblay-Turbiner-Winternitz potential; generalized harmonic oscillator; non-central potential; coherent state.

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  1. Calogero F., Solution of a three-body problem in one dimension, J. Math. Phys. 10 (1969), 2191-2197.
  2. Evans N.W., Group theory of the Smorodinsky-Winternitz system, J. Math. Phys. 32 (1991), 3369-3375.
  3. Fris J., Mandrosov V., Smorodinsky Ya.A., Uhlír M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
  4. Glauber R.J., Photon correlations, Phys. Rev. Lett. 10 (1963), 84-86.
  5. Glauber R.J., The quantum theory of optical coherence, Phys. Rev. 130 (1963), 2529-2539.
  6. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, Academic Press, New York, 1980.
  7. Kalnins E.G., Kress J.M., Miller W., Superintegrability and higher order integrals for quantum systems, J. Phys. A: Math. Theor. 43 (2010), 265205, 21 pages, arXiv:1002.2665.
  8. Kalnins E.G., Miller W., Pogosyan G.S., Superintegrability and higher order constants for classical and quantum systems, Phys. Atomic Nuclei 74 (2011), 914-918, arXiv:0912.2278.
  9. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  10. Quesne C., Superintegrability of the Tremblay-Turbiner-Winternitz quantum Hamiltonians on a plane for odd k, J. Phys. A: Math. Theor. 43 (2010), 082001, 10 pages, arXiv:0911.4404.
  11. Schrödinger E., Der stetige Übergang von der Mikro- zur Makromechanik, Naturwissenschaften 14 (1926), 664-666.
  12. Tremblay F., Turbiner A.V., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A: Math. Theor. 42 (2009), 242001, 10 pages, arXiv:0904.0738.
  13. Tempesta P., Turbiner A.V., Winternitz P., Exact solvability of superintegrable systems, J. Math. Phys. 42 (2001), 4248-4257, hep-th/0011209.
  14. Tremblay F., Turbiner A.V., Winternitz P., Periodic orbits for an infinite family of classical superintegrable systems, J. Phys. A: Math. Theor. 43 (2010), 015202, 14 pages, arXiv:0910.0299.
  15. Turbiner A., Hidden algebra of three-body integrable systems, Modern Phys. Lett. A 13 (1998), 1473-1483, solv-int/9805003.
  16. Unal N., Coherent states for Smorodinsky-Winternitz potentials, Cent. Eur. J. Phys. 7 (2009), 774-785.
  17. Unal N., Parametric time-coherent states for the hydrogen atom, Phys. Rev. A 63 (2001), 052105, 8 pages.
  18. Unal N., Parametric-time coherent states for Morse potential, Can. J. Phys. 80 (2002), 875-881.
  19. Unal N., Parametric-time coherent states for Smorodinsky-Winternitz potentials, J. Math. Phys. 48 (2007), 122107, 20 pages.
  20. Unal N., Parametric-time coherent states for the generalized MIC-Kepler system, J. Math. Phys. 47 (2006), 122105, 15 pages.
  21. Unal N., Path integration and coherent states for the 5D hydrogen atom, in Fluctuating Paths and Fields, Editors W. Janke, A. Pelster, H.J. Schmidt, M. Bachmann, World Sci. Publ., River Edge, NJ, 2001, 73-81.
  22. Unal N., Quasi-coherent states for harmonic oscillator with time-dependent parameters, J. Math. Phys. 53 (2012), 012102, 8 pages.
  23. Unal N., Smorodinsky-Winternitz potentials: coherent, state approach, Phys. Atomic Nuclei 74 (2011), 1758-1769.
  24. Watson G.N., A treatise on the theory of Bessel functions, Cambridge University Press, London, 1922.
  25. Wolfes J., On the three-body linear problem with three-body interaction, J. Math. Phys. 15 (1974), 1420-1424.

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