Let's consider the general solution of a particle in an infinite potential well with a cutoff c: $$\Psi(r,\theta,\phi) = \sum_{lm}a_lJ_l(kr)Y_l^m(\theta, \phi)$$ where $J_l$ is the Bessel function of the first kind and $J_l(kc) = 0$.
I read in a paper that the normalisation factor $a_l$ is given by $$\sqrt{\frac{2}{c^3J_{l+1}^2(k_n)}},$$ where $k_n$ is the n-th root of $J_l$ for a given l, that I deduce it comes from the condition $$\int_0^c|a_lJ_l(kr)|^2r^2dr=1$$ since we already know that the spherical harmonics are already normalised for the intergration over the solid angle. Does anyone know hot to derive the normalisation factor explicitly?
Thank in advance