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1: 15.18 Physical Applications
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)).
2: 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. …
3: 33.22 Particle Scattering and Atomic and Molecular Spectra
At positive energies E > 0 , ρ 0 , and: … R = m e c α 2 / ( 2 ) . … Both variable sets may be used for attractive and repulsive potentials: the ( ϵ , r ) set cannot be used for a zero potential because this would imply r = 0 for all s , and the ( η , ρ ) set cannot be used for zero energy E because this would imply ρ = 0 always. …
§33.22(vi) Solutions Inside the Turning Point
4: 14.31 Other Applications
§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)). …
5: 5.20 Physical Applications
Suppose the potential energy of a gas of n point charges with positions x 1 , x 2 , , x n and free to move on the infinite line - < x < , is given by
5.20.1 W = 1 2 = 1 n x 2 - 1 < j n ln | x - x j | .
5.20.2 P ( x 1 , , x n ) = C exp ( - W / ( k T ) ) ,
5.20.3 ψ n ( β ) = n e - β W d x = ( 2 π ) n / 2 β - ( n / 2 ) - ( β n ( n - 1 ) / 4 ) ( Γ ( 1 + 1 2 β ) ) - n j = 1 n Γ ( 1 + 1 2 j β ) .
5.20.4 W = - 1 < j n ln | e i θ - e i θ j | ,
6: 18.39 Physical Applications
Consider, for example, the one-dimensional form of this equation for a particle of mass m with potential energy V ( x ) : …
18.39.2 d 2 η d x 2 + 2 m 2 ( E - V ( x ) ) η = 0 .
For a harmonic oscillator, the potential energy is given by
18.39.3 V ( x ) = 1 2 m ω 2 x 2 ,
18.39.6 2 ψ + 2 m 2 ( E - V ( x ) ) ψ = 0 ,
7: 19.33 Triaxial Ellipsoids
§19.33(ii) Potential of a Charged Conducting Ellipsoid
The potential is
19.33.5 V ( λ ) = Q R F ( a 2 + λ , b 2 + λ , c 2 + λ ) ,
8: 29.19 Physical Applications
Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …
9: 12.17 Physical Applications
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).
10: 15.19 Methods of Computation
For fast computation of F ( a , b ; c ; z ) with a , b and c complex, and with application to Pöschl–Teller-Ginocchio potential wave functions, see Michel and Stoitsov (2008). …