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

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 $\frac{{d}^{2}\eta}{{dx}^{2}}+\frac{2m}{\hbar^{2}}\left(E-V(x)\right)\eta=0.$

For a harmonic oscillator, the potential energy is given by

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

where $\omega$ is the angular frequency. For (18.39.2) to have a nontrivial bounded solution in the interval $-\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 Encodings: TeX, pMML, png

The corresponding eigenfunctions are

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

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 $\nabla^{2}\psi+\frac{2m}{\hbar^{2}}\left(E-V(\mathbf{x})\right)\psi=0,$

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 ($\alpha=\beta=0$) 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).