Associated Legendre function: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Dan Nessett
(→‎Orthogonality relations: Redirected proof link to new proofs sub-page)
imported>Dan Nessett
(Moved reference to Bibliography sub-page. Moved external link to External Links sub-page)
Line 82: Line 82:
==Recurrence relations==
==Recurrence relations==


The functions satisfy the following difference equations, which are taken from Edmonds<ref>A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', Princeton University Press, 2nd edition (1960)</ref>
The functions satisfy the following difference equations, which are taken from Edmonds.


:<math>
:<math>
Line 112: Line 112:
=(\ell+m)P_{\ell-1}^{(m)}(x)-\ell x P_{\ell}^{(m)}(x)
=(\ell+m)P_{\ell-1}^{(m)}(x)-\ell x P_{\ell}^{(m)}(x)
</math><!-- Edmonds 2.5.25 -->
</math><!-- Edmonds 2.5.25 -->
==Reference==
<references />
==External link==
Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/LegendrePolynomial.html]

Revision as of 19:14, 1 September 2009

This article is developed but not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
Catalogs [?]
Proofs [?]
 
This editable, developed Main Article is subject to a disclaimer.

In mathematics and physics, an associated Legendre function Pl(m) is related to a Legendre polynomial Pl by the following equation

For even m the associated Legendre function is a polynomial, for odd m the function contains the factor (1-x ² )½ and hence is not a polynomial.

The associated Legendre functions are important in quantum mechanics and potential theory.

Differential equation

Define

where Pl(x) is a Legendre polynomial. Differentiating the Legendre differential equation:

m times gives an equation for Π(m)l

After substitution of

and after multiplying through with , we find the associated Legendre differential equation:

In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form

Extension to negative m

By the Rodrigues formula, one obtains

This equation allows extension of the range of m to: .

Since the associated Legendre equation is invariant under the substitution m → −m, the equations for Pl( ±m), resulting from this expression, are proportional.

To obtain the proportionality constant we consider

and we bring the factor (1−x²)m/2 to the other side. Equate the coefficient of the highest power of x on the left and right hand side of

and it follows that the proportionality constant is

so that the associated Legendre functions of same |m| are related to each other by

Note that the phase factor (−1)m arising in this expression is not due to some arbitrary phase convention, but arises from expansion of (1−x²)m.

Orthogonality relations

Important integral relations are

    [Proof]

Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds.