Answer to Question #125311 in Quantum Mechanics for Komal sardar

Question #125311
Discuss some applications of Legendre polynomial in physics. Derive in detail
Spherical harmonics Laguerre polynomials
1
Expert's answer
2020-07-06T17:06:53-0400

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.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS