§13.28 Physical Applications

Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.28

§13.28(i) Exact Solutions of the Wave Equation
§13.28(ii) Coulomb Functions
§13.28(iii) Other Applications

§13.28(i) Exact Solutions of the Wave Equation

Keywords:: Whittaker functions, confluent hypergeometric functions, coordinate systems, paraboloidal coordinates, quantum mechanics, wave equation
Permalink:: http://dlmf.nist.gov/13.28.i

The reduced wave equation $\nabla^{{2}}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi\eta}\mathop{\cos\/}\nolimits\phi$ , $y=2\sqrt{\xi\eta}\mathop{\sin\/}\nolimits\phi$ , $z=\xi-\eta$ , can be solved via separation of variables $w=f_{1}(\xi)f_{2}(\eta)e^{{ip\phi}}$ , where

13.28.1

$f_{1}(\xi)=\xi^{{-\frac{1}{2}}}V_{{\kappa,\frac{1}{2}p}}^{{(1)}}(2ik\xi)$ ,

$f_{2}(\eta)=\eta^{{-\frac{1}{2}}}V_{{\kappa,\frac{1}{2}p}}^{{(2)}}(-2ik\eta)$ ,

Symbols:: $k$ : wavenumber
Permalink:: http://dlmf.nist.gov/13.28.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

and $V^{{(j)}}_{{\kappa,\mu}}(z)$ , $j=1,2$ , denotes any pair of solutions of Whittaker’s equation (13.14.1). See Hochstadt (1971, Chapter 7).

For potentials in quantum mechanics that are solvable in terms of confluent hypergeometric functions see Negro et al. (2000).

§13.28(ii) Coulomb Functions

Keywords:: Coulomb functions, Whittaker functions
Permalink:: http://dlmf.nist.gov/13.28.ii

See Chapter 33.

§13.28(iii) Other Applications

Keywords:: coherent states, confluent hypergeometric functions, cosmology, many-body systems, tomography
Permalink:: http://dlmf.nist.gov/13.28.iii

For dynamics of many-body systems see Meden and Schönhammer (1992); for tomography see D’Ariano et al. (1994); for generalized coherent states see Barut and Girardello (1971); for relativistic cosmology see Crisóstomo et al. (2004).

§13.28 Physical Applications

Contents

§13.28(i) Exact Solutions of the Wave Equation

§13.28(ii) Coulomb Functions

§13.28(iii) Other Applications