Normalisation constant of the associated Laguerre polynomials

The normalisation constant ensures that the associated Laguerre polynomials are properly scaled, thereby maintaining the probabilistic interpretation of quantum states.

To determine the normalisation constant $N$ , we equate the coefficients of eq455 when $n=k$ to give

$\int_0^{\infty}x^me^{-x}L_k^m(x)L_k^m(x)dx=\frac{(m+k)!}{k!}\;\;\;\;\;\;\;\;456$

Substituting $L_k^m(x)=NL_k^m(x)$ and scaling the result of the integral to unity, we have $N^2\int_0^{\infty}x^me^{-x}L_k^m(x)L_k^m(x)dx=N^2\frac{(m+k)!}{k!}=1$ or

$N=\sqrt{\frac{k!}{(m+k)!}}$

Consider the integral of orthogonal associated Laguerre polynomials $\int_0^{\infty}\[x^me^{-x}L_k^m(x)L_{k+1}^m(x)\]dx=0$ . Substituting eq450 and rearranging gives