imported>Dan Nessett |
imported>Dan Nessett |
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
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.
-