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harmonic trapping potentials


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1: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
§14.30(i) Definitions
§14.30(iii) Sums
2: 12.17 Physical Applications
Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. … For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978). Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).
3: 17.17 Physical Applications
See Kassel (1995). … It involves q -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
4: Bibliography B
  • M. V. Berry (1966) Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.
  • A. Bhattacharjie and E. C. G. Sudarshan (1962) A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.
  • J. P. Boyd and A. Natarov (1998) A Sturm-Liouville eigenproblem of the fourth kind: A critical latitude with equatorial trapping. Stud. Appl. Math. 101 (4), pp. 433–455.
  • W. J. Braithwaite (1973) Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.
  • T. Busch, B. Englert, K. Rzażewski, and M. Wilkens (1998) Two cold atoms in a harmonic trap. Found. Phys. 28 (4), pp. 549–559.
  • 5: 18.39 Applications in the Physical Sciences
    argument a) The Harmonic Oscillator … The Schrödinger equation with potentialThis is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. … The eigenfunctions of L 2 are the spherical harmonics Y l , m l ( θ , ϕ ) with eigenvalues 2 l ( l + 1 ) , each with degeneracy 2 l + 1 as m l = l , l + 1 , , l . … …
    6: 18.38 Mathematical Applications
    Zonal Spherical Harmonics
    Ultraspherical polynomials are zonal spherical harmonics. … The solved Schrödinger equations of §18.39(i) involve shape invariant potentials, and thus are in the family of supersymmetric or SUSY potentials. SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …
    7: Donald St. P. Richards
    Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …
    8: 15.17 Mathematical Applications
    §15.17(iii) Group Representations
    For harmonic analysis it is more natural to represent hypergeometric functions as a Jacobi function (§15.9(ii)). …Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …
    9: Bibliography G
  • W. Gautschi (1974) A harmonic mean inequality for the gamma function. SIAM J. Math. Anal. 5 (2), pp. 278–281.
  • A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.
  • A. Gil and J. Segura (2000) Evaluation of toroidal harmonics. Comput. Phys. Comm. 124 (1), pp. 104–122.
  • A. Gil and J. Segura (2001) DTORH3 2.0: A new version of a computer program for the evaluation of toroidal harmonics. Comput. Phys. Comm. 139 (2), pp. 186–191.
  • D. Gómez-Ullate and R. Milson (2014) Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47 (1), pp. 015203, 26 pp..
  • 10: Bibliography H
  • R. L. Hall, N. Saad, and K. D. Sen (2010) Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.
  • D. R. Herrick and S. O’Connor (1998) Inverse virial symmetry of diatomic potential curves. J. Chem. Phys. 109 (1), pp. 11–19.
  • E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
  • L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.