§18.39 Physical Applications

Classical OP’s appear when the time-dependent Schrödinger equation is solved by separation of variables. Consider, for example, the one-dimensional form of this equation for a particle of mass $m$ with potential energy $V(x)$:

18.39.1		$$\left(\frac{-{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}+V(x)\right)\psi (x,t)=\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial t}\psi (x,t),$$
ⓘ Symbols: $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $m$: nonnegative integer, $x$: real variable, $V(x)$: potential energy and $\psi (x,t)$: wavefunction Permalink: http://dlmf.nist.gov/18.39.E1 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39 and Ch.18

where $\mathrm{\hslash }$ is the reduced Planck’s constant. On substituting $\psi (x,t)=\eta (x)\zeta (t)$, we obtain two ordinary differential equations, each of which involve the same constant $E$. The equation for $\eta (x)$ is

18.39.2		$$\frac{d^2\eta }{{dx}^2}+\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(x)\right)\eta =0.$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $m$: nonnegative integer, $x$: real variable, $V(x)$: potential energy and $E$: Energy Referenced by: §18.39(i) Permalink: http://dlmf.nist.gov/18.39.E2 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39 and Ch.18

For a harmonic oscillator, the potential energy is given by

18.39.3		$$V(x)=\frac{1}{2}m{\omega }^2{x}^2,$$
ⓘ Symbols: $m$: nonnegative integer, $x$: real variable, $V(x)$: potential energy and $\omega$: angular frequency Permalink: http://dlmf.nist.gov/18.39.E3 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39 and Ch.18

where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval , the constant $E$ (the total energy of the particle) must satisfy

18.39.4		$$E=E_n=\left(n+\frac{1}{2}\right)\mathrm{\hslash }\omega ,$$
$n=0,1,2,\mathrm{\dots }$.
ⓘ Defines: $E$: Energy (locally) Symbols: $n$: nonnegative integer and $\omega$: angular frequency Permalink: http://dlmf.nist.gov/18.39.E4 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39 and Ch.18

The corresponding eigenfunctions are

18.39.5		$${\eta }_n(x)={\pi }^{-\frac{1}{4}}{2}^{-\frac{1}{2}n}{(n!b)}^{-\frac{1}{2}}{H}_n\left(x/b\right){\mathrm{e}}^{-x^2/2b^2},$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.39.E5 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39 and Ch.18

where $b={(\mathrm{\hslash }/m\omega )}^{1/2}$, and $H_n$ is the Hermite polynomial. For further details, see Seaborn (1991, p. 224) or Nikiforov and Uvarov (1988, pp. 71-72).

A second example is provided by the three-dimensional time-independent Schrödinger equation

18.39.6		$${\nabla }^2\psi +\frac{2m}{{\mathrm{\hslash }}^2}\left(E-V(\mathbf{x})\right)\psi =0,$$
ⓘ Symbols: $m$: nonnegative integer, $V(x)$: potential energy, $\psi (x,t)$: wavefunction and $E$: Energy Permalink: http://dlmf.nist.gov/18.39.E6 Encodings: TeX, pMML, png See also: Annotations for §18.39(i), §18.39 and Ch.18

when this is solved by separation of variables in spherical coordinates (§1.5(ii)). The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. See Seaborn (1991, pp. 69-75).

For a third example, one in which the eigenfunctions are Laguerre polynomials, see Seaborn (1991, pp. 87-93) and Nikiforov and Uvarov (1988, pp. 76-80 and 320-323).

For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983).

For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials ($\alpha =\beta =0$) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974).

For physical applications of $q$-Laguerre polynomials see §17.17.

For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).