Associated Legendre function: Difference between revisions

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The associated Legendre functions are important in [[quantum mechanics]] and [[potential theory]].
The associated Legendre functions are important in [[quantum mechanics]] and [[potential theory]].
They are named after the French mathematician [[Adrien-Marie Legendre]] (1752–1833).


==Differential equation==
==Differential equation==

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In mathematics and physics, an associated Legendre function Plm 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. They are named after the French mathematician Adrien-Marie Legendre (1752–1833).

Differential equation

Define

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

m times gives an equation for Πml

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.[1]

Reference

  1. A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)