Question
I'm to show that <Pn|Pm> =nm in regards to Legendre Polynomials, suchas: n Pn(x) (2m+1)/2 0 1 1/2 1 x (3/2) * x 2 (3x2-1)/2 (3x2-1)/2 * (5/2) 3 (5x3-3x)/2 (5x3-3x)/2 * (7/2) with -1< x <1 . I believe the <Pn|Pm> notationdenotes "overlap" from the Born Rule, such that <1| 2> = As for the , all I have is<n|m> = nmfor notation. So it boils down to: i'm not even sure what the question isabout; my professor is very non-traditional and he threw this up ashomework without much explanation. The phrasing of the actualquestion is in the first line of this post. Any help at all will bevery very appreciated. Thanks. n Pn(x) (2m+1)/2 0 1 1/2 1 x (3/2) * x 2 (3x2-1)/2 (3x2-1)/2 * (5/2)
Explanation / Answer
This question is quite interesting! The point here is to show that the legendre polynomials areorthonormal to each other. I.e. = 0 if n != m and = 1 if n = m (not quite true, there should be ascaling factor of 2/(2n+1) in front of the kronecker delta) I myself do not know how to prove these two assertions in general.But it has been proved. Uhm, To do the first one, I would say look at the legendrepolynomials. If you multiply two of them together (not the same)then the product will be an odd function, so the integral will bezero. You can look here for further assistance maybe: http://en.wikipedia.org/wiki/Legendre_polynomial Especially the section on "Orthonormality".