bdforbes
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I'm looking right now at what purports to be the normalisation condition for the associated Laguerre polynomials:
[tex]\int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)dx=\frac{(n+k)!}{n!}\delta_{mn}[/tex]
However, in the context of Schroedinger's equation in spherical coordinates, I find that my normalisation integral has a different form:
[tex]|N|^2\int_0^\infty (\alpha r)^l e^{-\alpha r}[L_{n-l-1}^{2l+1}(\alpha r)]^2 r^2 dr=1[/tex]
I understand that I can evaluate this integral using the generating function of the associated Laguerre polynomials, but I'm a bit confused about why there are two forms for normalisation. Can anyone shed any light on this? Thanks.
[tex]\int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)dx=\frac{(n+k)!}{n!}\delta_{mn}[/tex]
However, in the context of Schroedinger's equation in spherical coordinates, I find that my normalisation integral has a different form:
[tex]|N|^2\int_0^\infty (\alpha r)^l e^{-\alpha r}[L_{n-l-1}^{2l+1}(\alpha r)]^2 r^2 dr=1[/tex]
I understand that I can evaluate this integral using the generating function of the associated Laguerre polynomials, but I'm a bit confused about why there are two forms for normalisation. Can anyone shed any light on this? Thanks.