Normalization of radial Laguerre-Gauss

[DivGradCurl]

Homework Statement

Normalization of radial Laguerre-Gauss

Normalize [tex] \Psi _n (r) = h_n L_n (2\pi r^2) e^{-\pi r^2} [/tex]

Homework Equations

[tex]\int _0 ^{\infty} e^{-x} \, x^k \, L_n ^{(k)} (x) \, L_m ^{(k)} (x) dx = \frac{(n+k)!}{n!} \delta _{m,n} [/tex]

The Attempt at a Solution

[tex]1 = \int _0 ^{\infty} \Psi _m ^{\ast} (r) \Psi _n (r) dr = \int _0 ^{\infty} h_m ^{\ast} L_m (2\pi r^2) e^{-\pi r^2} h_n L_n (2\pi r^2) e^{-\pi r^2} dr [/tex]

If I let [itex] x = 2\pi r^2[/itex], then I get [itex] dx = (4\pi r) dr[/itex]. The radial dependence bothers me. I think there's a step I'm missing out.

 

[DivGradCurl]

Just realized [itex]dx[/itex] is not an issue. I don't need to substitute it by a [itex]dr[/itex], so there's no problem. All I need to do is replace [itex]2\pi r^2[/itex] by [itex]x[/itex]. I'm done.