Normalization of radial Laguerre-Gauss

[DivGradCurl]

Homework Statement

Normalization of radial Laguerre-Gauss

Normalize $$\Psi _n (r) = h_n L_n (2\pi r^2) e^{-\pi r^2}$$

Homework Equations

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

The Attempt at a Solution

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

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

 

[DivGradCurl]

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