Legendre function

Legendre function is solution \(F\) of equation

\(\displaystyle  (1-x^2) F''(x) - 2xF'(x)+ \left( L  - \frac{m^2}{1-x^2} \right) F(x)=0 \)

where \(m\) and \(L\) are constant parameters. This equation comes from the Azimutal equation, that appears at the separation of variables for the Laplacian in spherical coordinates.

For the application to quantum mechanics of the hydrogen atom, it is assumed, that \(m\) is integer.

The solutions can be expressed in terms of polynomials, if \(L=\ell(\ell+1)\) for some integer \(\ell \ge |m|\); id est,

\(\displaystyle  (1\!-\!x^2) F''(x) - 2xF'(x)+ \left( \ell(\ell\!+\!1)  - \frac{m^2}{1\!-\!x^2} \right) F(x)=0 \)

The solution \(F(x)\) is denoted with

\(\mathrm{LegendreP} [\ell, m, x]=P_{\ell,m}(x)\)

Identifier LegendreP is recognised by the Mathematica software.

Case \(m=0\)
For \(m=0\), the equation becomes

\(\displaystyle  (1\!-\!x^2) F''(x) - 2xF'(x)+ \ell(\ell\!+\!1) F(x)=0 \)

For \(\ell\!=\!0\), the solution is constant.

For \(\ell\!=\!1\), the solution is linear function.

For \(\ell\!=\!1\), the solution \(F(x)\) is quadratic function, proportional to \(-1+3x^2\).

For integer non-negative \(\ell\), the solution is called LegendreP\(_\ell\), or "the Legendre polynomial" .

Another solution called LegendreQ\(_\ell\) is singular, it can be expressed through logarithms .

The only LegendreP is necessary for the direct application to the quantum mechanics with radial symmetry.

The polynomial LegendreP can be generated with the Rodriguez formula

\(\displaystyle \mathrm{LegendreP}_n(z) = \frac{1}{2^n n!} \frac{\mathrm d\, (z^2\!-\!1)^n } {\mathrm d z\, ^n}\)

Various \(m\)
For more general case, when \(m\) has no need to be zero, the solution can be expressed through the Legendre function, \(F(x)=\) LegendreP\(_{\ell,m}(x)\).

The Legendre function can be defined through the Legendre polynomial:

\(\displaystyle \mathrm{LegendreP}_{\ell,m}(x) = (-1)^m \Big(1\!-\!x^2\Big)^{m/2} \frac{\mathrm d\, \mathrm{LegendreP}_{\ell}(x)} {\mathrm d x\,^m} \)

Usually, namely this solution is called Legendre function.

If \(m\!=\!0\), then the Legendre function \(\mathrm{LegendreP}_{\ell,m}= \mathrm{LegendreP}_{\ell,0}= \mathrm{LegendreP}_{\ell} \) appears as polynomial of order \(\ell\), defined in the previous section.

Keywords
Atomic physics, Azimutal equation‎, Hydrogen wave function, Laplacian in spherical coordinates, Laplacian, Legendre function, Legendre polynomial, LegendreP, Molecular physics, Quantum mechanics, Schroedinger equation