12 Parabolic Cylinder Functions Applications 12.16 Mathematical Applications 12.18 Methods of Computation

§12.17 Physical Applications

ⓘ

Keywords:: Helmholtz equation, Laplacian, applications, boundary-value methods or problems, coordinate systems, harmonic trapping potentials, high-frequency scattering, parabolic cylinder, parabolic cylinder coordinates, parabolic cylinder functions, paraboloid of revolution, physical, quantum mechanics
Notes:: See Jeffreys and Jeffreys (1956, §§18.04 and 23.08) and Morse and Feshbach (1953a, b, pp. 553, 1403–1405).
Referenced by:: §1.5(ii)
Permalink:: http://dlmf.nist.gov/12.17
See also:: Annotations for Ch.12

The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation

12.17.1		$\nabla^{2}w+k^{2}w=0,$
ⓘ Symbols: $k$ : constant Referenced by: §12.17 Permalink: http://dlmf.nist.gov/12.17.E1 Encodings: TeX, pMML, png See also: Annotations for §12.17 and Ch.12

where $k$ is a constant, and $\nabla^{2}$ is the Laplacian

12.17.2		$\nabla^{2}=\frac{{\partial}^{2}}{{\partial x}^{2}}+\frac{{\partial}^{2}}{{% \partial y}^{2}}+\frac{{\partial}^{2}}{{\partial z}^{2}}$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $x$ : real variable, $y$ : real variable and $z$ : complex variable Permalink: http://dlmf.nist.gov/12.17.E2 Encodings: TeX, pMML, png See also: Annotations for §12.17 and Ch.12

in Cartesian coordinates $x, y, z$ of three-dimensional space (§1.5(ii)). By using instead coordinates of the parabolic cylinder $\xi,\eta,\zeta$ , defined by

12.17.3	$\displaystyle x$	$\displaystyle=\xi\eta,$
	$\displaystyle y$	$\displaystyle=\tfrac{1}{2}\xi^{2}-\tfrac{1}{2}\eta^{2},$
	$\displaystyle z$	$\displaystyle=\zeta,$
ⓘ Defines: $\xi$ : coordinate (locally), $\eta$ : coordinate (locally) and $\zeta$ : coordinate (locally) Symbols: $x$ : real variable, $y$ : real variable and $z$ : complex variable Permalink: http://dlmf.nist.gov/12.17.E3 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §12.17 and Ch.12

(12.17.1) becomes

12.17.4		$\frac{1}{\xi^{2}+\eta^{2}}\left(\frac{{\partial}^{2}w}{{\partial\xi}^{2}}+% \frac{{\partial}^{2}w}{{\partial\eta}^{2}}\right)+\frac{{\partial}^{2}w}{{% \partial\zeta}^{2}}+k^{2}w=0.$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $k$ : constant, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate Permalink: http://dlmf.nist.gov/12.17.E4 Encodings: TeX, pMML, png See also: Annotations for §12.17 and Ch.12

Setting $w=U(\xi)V(\eta)W(\zeta)$ and separating variables, we obtain

12.17.5	$\displaystyle\frac{{\mathrm{d}}^{2}U}{{\mathrm{d}\xi}^{2}}+\left(\sigma\xi^{2}% +\lambda\right)U$	$\displaystyle=0,$
	$\displaystyle\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}\eta}^{2}}+\left(\sigma\eta^{% 2}-\lambda\right)V$	$\displaystyle=0,$
	$\displaystyle\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\zeta}^{2}}+\left(k^{2}-% \sigma\right)W$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $W\left(\NVar{a},\NVar{x}\right)$ : parabolic cylinder function, $k$ : constant, $\xi$ : coordinate, $\eta$ : coordinate, $\zeta$ : coordinate, $\sigma$ : constant and $\lambda$ : constant Permalink: http://dlmf.nist.gov/12.17.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §12.17 and Ch.12

with arbitrary constants $\sigma,\lambda$ . The first two equations can be transformed into (12.2.2) or (12.2.3).

In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. See Buchholz (1969, §4) and Morse and Feshbach (1953a, pp. 515 and 553).

Buchholz (1969) collects many results on boundary-value problems involving PCFs. Miller (1974) treats separation of variables by group theoretic methods. Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator.

Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. 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).

12.16 Mathematical Applications 12.18 Methods of Computation

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§12.17 Physical Applications

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