Question

Discuss some applications of Legendre polynomial in physics. Derive in detail
Spherical harmonics Laguerre polynomials

Answer 1

In physics, we want to solve often the problem:

$\Delta\phi(\vec r) = cf(\vec r)$

The differential operator $\Delta$ we can write in different coordinate systems. For exapmle, in spherical coordinate system:

$\Delta_{r\phi\theta} g(r,\phi,\theta) = \frac{1}{r}\frac{\partial}{\partial r}\left(\frac{1}{r}\frac{\partial}{\partial r}\right)g(r,\phi,\theta) +\frac{1}{r^2}\Delta_{\phi\theta}g(r,\phi,\theta)$

If function (the angular part of g) $\psi(\phi,\theta)$ is eigenfunction for a operator $\Delta_{\phi\theta}$ then such function equals $P^l_m(cos(\phi))$ - Legendre polinomials.

From radial part of this problem we can obtain the eigenvalue problem for the $R(r)$ - radial part of the function g. The eigenfunctions of this problems will be Laguerre polynomials.