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… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).
… ►See Kassel (1995). … ►It involves -generalizations of exponentials and Laguerre polynomials, and has been applied to the problems of the harmonic oscillator and Coulomb potentials. …
… ►At positive energies , , and: … ► . … ►Both variable sets may be used for attractive and repulsive potentials: the set cannot be used for a zero potential because this would imply for all , and the set cannot be used for zero energy because this would imply always. … ►
§33.22(vi) Solutions Inside the Turning Point…
§14.31(ii) Conical Functions… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …
… ►Suppose the potential energy of a gas of point charges with positions and free to move on the infinite line , is given by ►
… ►Consider, for example, the one-dimensional form of this equation for a particle of mass with potential energy : … ►
18.39.2►For a harmonic oscillator, the potential energy is given by ►
… ►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …
… ►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).