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

SIGMA 8 (2012), 074, 16 pages      arXiv:1210.4632      https://doi.org/10.3842/SIGMA.2012.074
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Ladder Operators for Lamé Spheroconal Harmonic Polynomials

Ricardo Méndez-Fragoso a and Eugenio Ley-Koo b
a) Facultad de Ciencias, Universidad Nacional Autónoma de México, México
b) Instituto de Física, Universidad Nacional Autónoma de México, México

Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012

Abstract
Three sets of ladder operators in spheroconal coordinates and their respective actions on Lamé spheroconal harmonic polynomials are presented in this article. The polynomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lamé polynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers $n_1$ and $n_2$ counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components $\hat L_x$, $\hat L_y$, $\hat L_z$ of the angular momentum connect pairs of the four species of polynomials of a chosen kind and angular momentum. The third set of operators, the cartesian components $\hat p_x$, $\hat p_y$, $\hat p_z$ of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. Relationships among spheroconal harmonics at the levels of the three sets of operators are illustrated.

Key words: Lamé polynomials; spheroconal harmonics; ladder operators.

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References

  1. Abramowitz M., Stegun I.A., Handbook of mathematical functions, Dover, New York, 1965.
  2. Aquilanti V., Caligiana A., Cavalli S., Hydrogenic elliptic orbitals, Coulomb Sturmian sets, and recoupling coefficients among alternative bases, Int. J. Quantum Chem. 92 (2003), 99-117.
  3. Aquilanti V., Caligiana A., Cavalli S., Coletti C., Hydrogenic orbitals in momentum space and hyperspherical harmonics: elliptic Sturmian basis sets, Int. J. Quantum Chem. 92 (2003), 212-228.
  4. Aquilanti V., Tonzani S., Three-body problem in quantum mechanics: hyperspherical elliptic coordinates and harmonic basis sets, J. Chem. Phys. 120 (2004), 4066-4073.
  5. Grosche C., Karayan K.H., Pogosyan G.S., Sissakian A.N., Quantum motion on the three-dimensional sphere: the ellipso-cylindrical bases, J. Phys. A: Math. Gen. 30 (1997), 1629-1657.
  6. Kramers H.A., Ittmann G.P., Zur Quantelung des asymmetrischen Kreisels, Z. Phys. 53 (1929), 553-565.
  7. Kronig R. de L., Rabi I.I., The symmetrical top in the undulatory mechanics, Phys. Rev. 29 (1927), 262-269.
  8. Kroto H.W., Molecular rotation spectra, John Wiley & Sons, London, 1975.
  9. Ley-Koo E., Méndez-Fragoso R., Properties of the spectra of asymmetric molecules: matrix evaluation in bases of spherical harmonics and common generating function, Rev. Mexicana Fís. 54 (2008), 69-77.
  10. Ley-Koo E., Méndez-Fragoso R., Rotational states of asymmetric molecules revisited: matrix evaluation and generating function of Lamé functions, Rev. Mexicana Fís. 54 (2008), 162-172.
  11. Ley-Koo E., Sun G.H., Ladder operators for quantum systems confined by dihedral angles, SIGMA 8 (2012), 060, 15 pages, arXiv:1209.2497.
  12. Liu Q.H., Xun D.M., Shan L., Raising and lowering operators for orbital angular momentum quantum numbers, Internat. J. Theoret. Phys. 49 (2010), 2164-2171.
  13. Lukach I., A complete set of quantum-mechanical observables on a two-dimensional sphere, Theoret. and Math. Phys. 14 (1973), 271-281.
  14. Lukach I., Smorodinski Ya.A., Separation of variables in a spheroconical coordinate system and the Schrödinger equation for a case of noncentral forces, Theoret. and Math. Phys.
  15. Lukach I., Smorodinski Ya.A., The wave functions of an asymmetrical top, Soviet Phys. JETP 30 (1970), 728-730.
  16. Lütgemeier F., Zur Quantentheorie des drei- und mehratomigen Moleküls, Z. Phys. 38 (1926), 251-263.
  17. Méndez-Fragoso R., Ley-Koo E., Lamé spheroconal harmonics in atoms and molecules, Int. J. Quantum Chem. 110 (2010), 2765-2774.
  18. Méndez-Fragoso R., Ley-Koo E., Rotations of asymmetric molecules and the hydrogen atom in free and confined congurations, Adv. Quantum Chem. 62 (2011), 137-213.
  19. Méndez-Fragoso R., Ley-Koo E., The hydrogen atom in a semi-infinite space with an elliptical cone boundary, Int. J. Quantum Chem. 111 (2011), 2882-2897.
  20. Morse P.M., Feshbach H., Methods of theoretical physics, Vols. 1, 2, McGraw-Hill Book Co. Inc., New York, 1953.
  21. Niven W.D., On ellipsoidal harmonics, Philos. Trans. R. Soc. Lond. Ser. A 182 (1891), 231-278.
  22. Odake S., Sasaki R., Discrete quantum mechanics, J. Phys. A: Math. Theor. 44 (2011), 353001, 47 pages, arXiv:1104.0473.
  23. Odake S., Sasaki R., Orthogonal polynomials from Hermitian matrices, J. Math. Phys. 49 (2008), 053503, 43 pages, arXiv:0712.4106.
  24. Patera J., Winternitz P., A new basis for the representations of the rotation group. Lamé and Heun polynomials, J. Math. Phys. 14 (1973), 1130-1139.
  25. Patera J., Winternitz P., On bases for irreducible representations of O(3) suitable for systems with an arbitrary finite symmetry group, J. Chem. Phys. 65 (1976), 2725-2731.
  26. Piña E., Algunas propiedades de los operadores de escalera, Rev. Mexicana Fís. 41 (1995), 913-924.
  27. Piña E., Some properties of the spectra of asymmetric molecules, J. Mol. Structure: THEOCHEM 493 (1999), 159-170.
  28. Piña E., Jiménez-Lara L., Properties of new coordinates for the general three-body problem, Celestial Mech. Dynam. Astronom. 82 (2002), 1-18.
  29. Reiche F., Rademacher H., Die Quantelung des symmetrischen Kreisels nach Schrödingers Undulationsmechanik, Z. Phys. 39 (1926), 444-464.
  30. Sun G.H., Dong S.H., New type shift operators for circular well potential in two dimensions, Phys. Lett. A 374 (2010), 4112-4114.
  31. Sun G.H., Dong S.H., New type shift operators for three-dimensional infinite well potential, Modern Phys. Lett. A 26 (2011), 351-358.
  32. Valdéz M.T., Piña E., The rotational spectra of the most asymmetric molecules, Rev. Mexicana Fís. 52 (2006), 220-229.
  33. Volkmer H., Lamé functions, in NIST Handbook of Mathematical Functions, Editors F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, U.S. Dept. Commerce, Washington, DC, 2010, 683-695.
  34. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  35. Witmer E.E., The rotational energy of the polyatomic molecule as an explicit function of the quantum numbers, Proc. Natl. Acad. Sci. USA 12 (1926), 602-608.

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