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

§12.17 Physical Applications

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

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:: : constant
Referenced by:: §12.17
Permalink:: http://dlmf.nist.gov/12.17.E1
Encodings:: TeX, pMML, png

where 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:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : real variable, : real variable and : complex variable
Permalink:: http://dlmf.nist.gov/12.17.E2
Encodings:: TeX, pMML, png

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

12.17.3

$x=\xi\eta,$

$y=\tfrac{1}{2}\xi^{2}-\tfrac{1}{2}\eta^{2},$

$z=\zeta,$

Symbols:: : real variable, : real variable, : complex variable, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate
Permalink:: http://dlmf.nist.gov/12.17.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

(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:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : constant, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate
Permalink:: http://dlmf.nist.gov/12.17.E4
Encodings:: TeX, pMML, png

Setting $w=\mathop{U\/}\nolimits(\xi)\mathop{V\/}\nolimits(\eta)\mathop{W\/}\nolimits(\zeta)$ and separating variables, we obtain

12.17.5

$\frac{{d}^{2}\mathop{U\/}\nolimits}{{d\xi}^{2}}+\left(\sigma\xi^{2}+\lambda% \right)\mathop{U\/}\nolimits=0,$

$\frac{{d}^{2}\mathop{V\/}\nolimits}{{d\eta}^{2}}+\left(\sigma\eta^{2}-\lambda% \right)\mathop{V\/}\nolimits=0,$

$\frac{{d}^{2}\mathop{W\/}\nolimits}{{d\zeta}^{2}}+\left(k^{2}-\sigma\right)% \mathop{W\/}\nolimits=0,$

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 12.16 Mathematical Applications 12.18 Methods of Computation