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 $V\left(x\right)$:

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

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

18.39.2 | $$\frac{{\mathrm{d}}^{2}\eta}{{\mathrm{d}x}^{2}}+\frac{2m}{{\mathrm{\hslash}}^{2}}\left(E-V\left(x\right)\right)\eta =0.$$ | ||

For a harmonic oscillator, the potential energy is given by

18.39.3 | $$V\left(x\right)=\frac{1}{2}m{\omega}^{2}{x}^{2},$$ | ||

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 | $$E={E}_{n}=\left(n+\frac{1}{2}\right)\mathrm{\hslash}\omega ,$$ | ||

$n=0,1,2,\mathrm{\dots}$. | |||

The corresponding eigenfunctions are

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

where $b={\left(\mathrm{\hslash}/m\omega \right)}^{1/2}$, 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 | $${\mathrm{\nabla}}^{2}\psi +\frac{2m}{{\mathrm{\hslash}}^{2}}\left(E-V\left(\mathbf{x}\right)\right)\psi =0,$$ | ||

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 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.