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 with potential energy :
where is the reduced Planck’s constant. On substituting , we obtain two ordinary differential equations, each of which involve the same constant . The equation for is
For a harmonic oscillator, the potential energy is given by
where is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval , the constant (the total energy of the particle) must satisfy
The corresponding eigenfunctions are
A second example is provided by the three-dimensional time-independent Schrödinger equation
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 Szegő–Szász inequality (18.14.20) for Legendre polynomials () 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 -Laguerre polynomials see §17.17.