13 Legendre Polynomials
13.1 Definition of Legendre Polynomials
Legendre polynomials \(P_{n}\left(x\right)\) are defined over the Hilbert space \(\mathcal{L}^{2}[-1,1]\) (an \(\mathcal{L}^{2}-\)space defined over the interval \(\left[-1,1\right]\in\mathbb{R}\)) with weight function \(w\left(x\right)=1\). This is typically denoted as the ordered pair \(\mathcal{H}=\left(\mathcal{L}^{2}[-1,1],w\left(x\right)=1\right)\). This means that the inner product between two functions is
\[\langle f|g\rangle\triangleq\int^{1}_{-1}f\left(x\right)^{*}g\left(x\right)w\left(x\right)\text{d}x\]
where \(w\left(x\right)=1\). Hence, it is the usual inner product for functions.
- The Legendre polynomials are solutions to Legendre’s differential equation
\[\left(1-x^{2}\right)y''-2xy'+n\left(n+1\right)y=0.\]
- First few terms:
\[\begin{aligned} P_{0}\left(x\right) & =1\\ P_{1}\left(x\right) & =x\\ P_{2}\left(x\right) & =\frac{1}{2}\left(3x^{2}-1\right) \end{aligned}\]
- Rodrigues’ formula:
\[P_{n}\left(x\right)=\frac{1}{2^{n}n!}\text{d}^{n}_{x}\left(x^{2}-1\right)^{n}\]
- Recursion formula (Bonnet’s formula): given \(P_{0}\), \(P_{1}\),
\[\left(n+1\right)P_{n+1}\left(x\right)=\left(2n+1\right)xP_{n}\left(x\right)-nP_{n-1}\left(x\right)\]
- Generating function
\[\left(1-2xt+t^{2}\right)^{-1/2}=\sum^{\infty}_{n=0}P_{n}\left(x\right)t^{n}\]
meaning \[P_{n}\left(x\right)=\left.\partial^{n}_{t}\left(1-2xt+t^{2}\right)^{-1/2}\right|_{t=0}\]
- Contour integral expression
\[P_{n}\left(z\right)=\frac{1}{2\pi i}\oint_{C}\left(1-2tz+t^{2}\right)^{-1/2}t^{-n-1}\text{d}t\]
where \(C\) encircles the origin anticlockwise.
13.2 Orthogonality & Normalisation
\[\int^{1}_{-1}P_{n}\left(x\right)P_{m}\left(x\right)\text{d}x=\frac{2}{2n+1}\delta_{nm}.\]
13.3 Completeness
The Legendre Polynomials form an orthonormal basis for functions on the interval \(\left[-1,1\right]\). To prove that they form a complete orthonormal basis is tricky. The proof is not covered here. The statement, though, is as follows.
Given any piecewise continuous function \(f\left(x\right)\) with at most finitely many discontinuities on the interval \(\left[-1,1\right]\), the sequence of partial sums
\[f_{n}\left(x\right)=\sum^{n}_{l=0}a_{l}P_{l}\left(x\right)\label{eq:f =00003D sum Pl}\]
converges to \(f\left(x\right)\) in the limit \(n\rightarrow\infty\). Here,
\[a_{l}=\frac{2l+1}{2}\int^{1}_{-1}f\left(x\right)P_{l}\left(x\right)\text{d}x.\]
Another way to state this is
\[\sum^{\aleph_{0}}_{l=0}\frac{2l+1}{2}P_{l}\left(x\right)P_{l}\left(y\right)=\delta\left(x-y\right)\]
with \(x,y\in\left[-1,1\right]\). This is a statement that we can resolve the identity into the basis of \(|P_{l}\rangle.\) Hence, The Legendre Polynomials form a complete basis for the Hilbert space \(\mathcal{H}=\left(\mathcal{L}^{2}\left[-1,1\right],w\left(x\right)=1\right)\).
13.4 QM example: spherical potential, angular part
If you are interested to see where Legendre Polynomials appear in quantum mechanics, please see my notes on that subject [https://www.felixflicker.com/files/PX2132/PX2132_notes_2022.pdf].