§18.39 Physical Applications

Keywords:: classical orthogonal polynomials
Referenced by:: §14.31(iii)
Permalink:: http://dlmf.nist.gov/18.39

§18.39(i) Quantum Mechanics
§18.39(ii) Other Applications

§18.39(i) Quantum Mechanics

Keywords:: Hermite polynomials, Laguerre polynomials, Legendre polynomials, Schrödinger equation, classical orthogonal polynomials, differential equations, harmonic oscillators, partial differential equations, quantum mechanics, reduced Planck’s constant
Permalink:: http://dlmf.nist.gov/18.39.i

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 $\left(\frac{-\hbar^{2}}{2m}\frac{{\partial}^{2}}{{\partial x}^{2}}+V(x)\right)\psi(x,t)=i\hbar\frac{\partial}{\partial t}\psi(x,t),$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $m$ : nonnegative integer, $x$ : real variable, $V(x)$ : potential energy and $\psi(x,t)$ : wavefunction
Permalink:: http://dlmf.nist.gov/18.39.E1
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where $\hbar$ 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 $\frac{{d}^{2}\eta}{{dx}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x)\right)\eta=0.$

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
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For a harmonic oscillator, the potential energy is given by

18.39.3 $V(x)=\tfrac{1}{2}m\omega^{2}x^{2},$

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

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

18.39.4 $E=E_{n}=\left(n+\tfrac{1}{2}\right)\hbar\omega,$ $n=0,1,2,\dots$ .

Symbols:: $n$ : nonnegative integer, $\omega$ : angular frequency and $E$ : Energy
Permalink:: http://dlmf.nist.gov/18.39.E4
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The corresponding eigenfunctions are

18.39.5 $\eta _{n}(x)=\pi^{{-\frac{1}{4}}}2^{{-\frac{1}{2}n}}(n!\, b)^{{-\frac{1}{2}}}\mathop{H_{{n}}\/}\nolimits\!\left(x/b\right)e^{{-x^{2}/2b^{2}}},$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $e$ : base of exponential function, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.39.E5
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where $b=(\hbar/m\omega)^{{1/2}}$ , and $\mathop{H_{{n}}\/}\nolimits$ 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 $\nabla^{2}\psi+\frac{2m}{\hbar^{2}}\left(E-V(\mathbf{x})\right)\psi=0,$

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

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).

§18.39(ii) Other Applications

Keywords:: Legendre polynomials, electrostatics, fluid dynamics
Permalink:: http://dlmf.nist.gov/18.39.ii

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 Szegö–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).

§18.39 Physical Applications

Contents

§18.39(i) Quantum Mechanics

§18.39(ii) Other Applications