Laguerre polynomials

Laguerre polynomials (named after Edmond Laguerre ) are the solutions of the Laguerre differential equation

The -th Laguerre polynomial can be on the Rodrigues formula

Represent. It is a polynomial of degree. Over the first Laguerre polynomials

Can be further calculate the following recursion:

In physics, typically a definition of the Laguerre polynomial is used, which is larger by a factor.

Associated Laguerre polynomials

The associated Laguerre polynomials are related to the ordinary Laguerre polynomials over

Together. Your Rodrigues formula is

The associated Laguerre polynomials satisfy the associated Laguerre equation

Are the first associated Laguerre polynomials

Hydrogen atom

The Laguerre polynomials have an application in quantum mechanics in solving the Schrödinger equation for the hydrogen atom and in the general case of a Coulomb potential. By means of the associated Laguerre polynomials can write the radial part of the wave function as

( Normalization constant, characteristic length, principal quantum number, the orbital angular momentum quantum number ). The associated Laguerre polynomials thus have a crucial role here. The normalized total wave function is characterized by

Given, with the principal quantum number, the orbital angular momentum quantum number, magnetic quantum number, Bohr radius and atomic number. The function of the associated Laguerre polynomials, and are the spherical harmonics.