Chemistry - Associated Legendre polynomials

The associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by:

$P_\ell ^m (x) = \frac{(-1)^m}{2^\ell \ell!} (1 - x^2) ^ \frac{m}{2} \left( \frac{d}{dx} \right ) ^{\ell+m} (x^2 - 1)^\ell.$

These differ from the Legendre polynomials.

They satisfy the orthogonality condition

$\int_{-1}^{1} P_k ^m P_\ell ^m dx = \delta _{k,\ell} \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}$

Where $\delta _{k,\ell}$ is the Kronecker delta.

In many occasions in physics, Legendre polynomials occur where spherical symmetry is involved. Then, the Legendre polynomials represent the dependence in azimuth as $\ x = \cos\theta$ . The associated Legendre polynomials are extensions of the above, where the $\sqrt{1-x^2}$ terms are $\ \sin\theta$ . In other words, while the Legendre polynomials are polynomials of $\ \cos\theta$ , and the Associated Legendre polynomials are polynomials of $\ \cos\theta$ and $\ \sin\theta$ .

The Associated Legendre polynomials are an important part of the spherical harmonics.

The Legendre polynomials are closely related to hypergeometric series. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3). As such, Legendre polynomials can be generalized to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces.

References

A.R. Edmonds, Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9 See chapter 2.
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 521-09209-4 See chapter 3

Categories: Polynomials | Special functions | Quantum mechanics

Associated Legendre polynomials

See also

References