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11: 13.28 Physical Applications
For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000). …
12: 16.24 Physical Applications
They are also potentially useful for the solution of more complicated restricted lattice walk problems, and the 3D Ising model; see Barber and Ninham (1970, pp. 147–148). …
13: 19.18 Derivatives and Differential Equations
The next four differential equations apply to the complete case of R F and R G in the form R - a ( 1 2 , 1 2 ; z 1 , z 2 ) (see (19.16.20) and (19.16.23)). … Similarly, the function u = R - a ( 1 2 , 1 2 ; x + i y , x - i y ) satisfies an equation of axially symmetric potential theory: …
14: 9.16 Physical Applications
Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …
15: 22.19 Physical Applications
22.19.4 d 2 x ( t ) d t 2 = - d V ( x ) d x ,
where V ( x ) is the potential energy, and x ( t ) is the coordinate as a function of time t . The potential
22.19.5 V ( x ) = ± 1 2 x 2 ± 1 4 β x 4
16: 10.73 Physical Applications
Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …
17: 23.21 Physical Applications
In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form ( 1 - x 2 ) ( 1 - k 2 x 2 ) . The Weierstrass function plays a similar role for cubic potentials in canonical form g 3 + g 2 x - 4 x 3 . …
18: DLMF Project News
error generating summary
19: 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.
  • 20: Bibliography W
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.