Talk:Associated Legendre function/Addendum

I added a proof of orthogonality and a derivation of the normalization constant for the first equation in the Orthogonality relations section in on the main page. Dan Nessett 16:42, 11 July 2009 (UTC)

Comments on proof

1. The proof starts out by implicitly proving the anti-Hermiticity of

$\nabla _{x}\equiv {\frac {d}{dx}}.$

Indeed, let w(x) be a function with w(1) = w(−1) = 0, then

$\langle wg|\nabla _{x}f\rangle =\int _{-1}^{1}w(x)g(x)\nabla _{x}f(x)dx=\left[w(x)g(x)f(x)\right]_{-1}^{1}-\int _{-1}^{1}{\Big (}\nabla _{x}w(x)g(x){\Big )}f(x)dx=-\langle \nabla _{x}(wg)|f\rangle$

Hence

$\nabla _{x}^{\dagger }=-\nabla _{x}\;\Longrightarrow \;\left(\nabla _{x}^{\dagger }\right)^{l+m}=(-1)^{l+m}\nabla _{x}^{l+m}$

The latter result is used in the proof given in the Addendum.

2. When as an intermediate the ordinary Legendre polynomials P_l are introduced, we may use a result from the theory of orthogonal polynomials. Namely, a Legendre polynomial of order l is orthogonal to any polynomial of lower order. We meet (k ≤ l)

$\langle w\nabla _{x}^{m}P_{k}|\nabla _{x}^{m}P_{l}\rangle \quad {\hbox{with}}\quad w\equiv (1-x^{2})^{m},$

then

$\langle w\nabla _{x}^{m}P_{k}|\nabla _{x}^{m}P_{l}\rangle =(-1)^{m}\langle \nabla _{x}^{m}(w\nabla _{x}^{m}P_{k})|P_{l}\rangle$

The bra is a polynomial of order k, and since k ≤ l, the bracket is non-zero only if k = l.

Then, knowing this, the hard work (given in the Addendum) of computing the normalization constant remains.

--Paul Wormer 15:13, 12 July 2009 (UTC)

While there are probably many ways to demonstrate the orthogonality property of Associated Legendre Functions, recasting the proof in terms of the anti-Hermiticity of the differentiation operator assumes the reader is familiar with operator mathematics. The proof as it stands has the advantage that it only requires the reader to understand basic calculus. As I have commented in the forum thread (Citizendium Forums > Workgroup Boards > Physics (Moderator: salsb) > Best way to incorporate orthogonality/orthonormality proofs) that precedes this talk conversation, there is a common pedagogical approach that uses operators as a notational convenience rather than as first-class objects. Since this proof should be accessible to a wide audience, I would argue that we should keep it as it is.

Nevertheless, the points you provide are valid and I have no objection to incorporating them in the proof, perhaps as comments or as integrated parenthetical text. The only objection I would have is if the proof becomes incomprehensible to those who do not understand operator mathematics (e.g., the concept of operators as infinite dimensional matrices). Dan Nessett 14:50, 13 July 2009 (UTC)

I integrated the comments made by Paul Wormer into the proof. They stand in a comments section below the body of the proof. Dan Nessett 15:10, 14 July 2009 (UTC)