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18 Orthogonal PolynomialsApplications

§18.39 Physical Applications

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§18.39(i) Quantum Mechanics

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 (-22m2x2+V(x))ψ(x,t)=itψ(x,t),

where is the reduced Planck’s constant. On substituting ψ(x,t)=η(x)ζ(t), we obtain two ordinary differential equations, each of which involve the same constant E. The equation for η(x) is

18.39.2 d2ηdx2+2m2(E-V(x))η=0.

For a harmonic oscillator, the potential energy is given by

18.39.3 V(x)=12mω2x2,

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

18.39.4 E=En=(n+12)ω,
n=0,1,2,.

The corresponding eigenfunctions are

18.39.5 ηn(x)=π-142-12n(n!b)-12Hn(x/b)e-x2/2b2,

where b=(/mω)1/2, and Hn 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 2ψ+2m2(E-V(x))ψ=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 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

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 (α=β=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).