/sci/ - Science & Math

>>12419928
Yes I know that that is 1 by definition, I just don't get how it pops out in this case

>>12420188
>the generalized laguerre polynomials require two numbers
But I wrote them with two numbers, the subscript which is [math]n-l-1[/math] and the superscript which is [math]2l+1[/math], right?
>as it stands your first line should equal 0 as your spherical harmonics would be orthogonal
Sorry I think I dismissed something important, there was a [math]\hat{z}=cos\theta[/math] all along, like this:
[math]\langle 310 |E\hat{z}| 320 \rangle=E\int_{0}^{2\pi} d\phi\int_{0}^{\pi}sin\theta \, d\theta \, (Y_{1}^{0})^* \,cos\theta \, Y_{2}^{0} \langle 3 | 3 \rangle[/math]
pic related is what I'm following (https://youtu.be/omX2cirpXqk?t=360))

>>12420206
>only between different n and same l polynomials
Yes you're right. Then IDK
>the orthogonality comes from the spherical harmonics
Wait what? I thought those were two indipendent orthogonalities. Spherical harmonics don't even appear in the Laguerre polynomials