Derivation process of Selection Rule of hydrogenic atom

[smjchris]

Homework Statement

Prove that the radial integral is always non-zero

Relevant Equations

Selection rule, wavefunction of hydrogeninc atom.

This page is Quantum mechanics by bransden. My homework is explain why there is no regulation of quantum number n in selection rule. Also explain that by solving that integral of radial part is always non-zero.

∫∞0[rRnl(r)]Rn′l′(r)r2dr

(n is different with n')

I tried to solve it by just calculate it, but I can't calculate the associated laguerre polynomial. I think the answer is using orthogonality, but how can I solve it?

 

[kuruman]

Also explain that by solving that integral of radial part is always non-zero.

I think always is too strong. Consider two wavefunctions with different $n$ but the same $l$ and $m$ quantum numbers. What should the following integral give you?$$I=\int_0^{\infty}R_{n'l}R_{nl}~r^2dr\int_0^{2\pi}d\phi\int_0^{\pi}Y^*_{lm}Y_{lm}~\sin\theta d\theta.$$