Section 5.2 Legendre Polynomials
Legendre polynomials are special solutions of Legendre’s equation, i.e. the differential equation
\begin{equation}
\frac{d^2 P}{d z^2}
- \frac{2z}{1-z^2}\frac{d P}{d z}
+ \frac{\ell(\ell+1)}{1-z^2} P
= 0\tag{5.2.1}
\end{equation}
for \(\ell\) a constant, non-negative integer.
The first few Legendre polynomials are:
\begin{align}
P_0(z) \amp= 1\tag{5.2.2}\\
P_1(z) \amp= z\tag{5.2.3}\\
P_2(z) \amp= \frac12 \, (3z^2-1)\tag{5.2.4}\\
P_3(z) \amp= \frac12 \, (5z^3-3z)\tag{5.2.5}\\
P_4(z) \amp= \frac18 \, (35z^4-30z^2+3)\tag{5.2.6}\\
P_5(z) \amp= \frac18 \, (63z^5-70z^3+15z)\tag{5.2.7}
\end{align}
You can use the Sage code below to display the algebraic expression for the Legendre polynomial \(P_n(x)\) for any integer value of \(n\text{,}\) and then its graph.
Below is a list of several useful patterns to the Legendre polynomials. Most of them can be proven using Rodrigues’ formula, (5.3.1) in Section 5.3.
- \(P_\ell(z)\) is a polynomial of degree \(\ell\text{.}\)
- Each \(P_\ell(z)\) contains only odd or only even powers of \(z\text{,}\) depending on whether \(\ell\) is even or odd. Therefore, each \(P_\ell(z)\) is either an even or an odd function.
- The overall coefficient for each solution is conventionally chosen so that \(P_\ell(1)=1\text{.}\) As discussed in Section 5.4, this is an inconvenient convention that we are stuck with!
- Since the differential operator in (7.2.1) is Hermitian (unproven), we are guaranteed by a deep theorem of mathematics (Sturm-Liouville theory, see Section 5.1) that the Legendre polynomials are orthogonal for different values of \(\ell\) (just as with Fourier series) i.e.\begin{equation} \int\limits_{-1}^1 P^*_k(z) \, P_\ell(z) \,dz = \frac{\delta_{k\ell}}{\ell+\frac12}\tag{5.2.8} \end{equation}The “squared norm” of \(P_\ell\) is just \(1/(\ell+\frac12)\text{.}\) To normalize each \(P_\ell(z)\) for quantum mechanics, it should be multiplied by \(\sqrt{\ell+\frac12}\text{.}\)
- The Legendre polynomials can also be defined, recursively, from two initial values and a recursion relation, i.e.\begin{align} P_0(x) \amp= 1,\notag\\ P_1(x) \amp= x,\notag\\ (n+1)P_{n+1}(x) \amp= (2n+1)x P_n(x) - nP_{n-1}(x)\tag{5.2.9} \end{align}
Notice that Legendre’s equation (5.2.1) is a different equation for different values of \(\ell\text{.}\) For a given value of \(\ell\text{,}\) you should expect two solutions of (5.2.1). We have only given one for the special case \(\ell\in \{0, 1, 2, 3, \dots\}\text{.}\) It turns out that the other solution for these values of \(\ell\) and both solutions for other values \(\ell\) are not regular (i.e. blow up) at \(z=\pm 1\text{.}\) We can often discard these irregular solutions as unphysical for the problems we are solving.
