Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 2 / ii / Solution

Separate the interior Helmholtz equation in the same angular basis. Each radial coefficient $F_{\ell }(r)$ satisfiesFor $s>0$, set $x=\sqrt{s}r$. The equation becomes $x^2{f}_{\ell }^{\prime \prime }+2x{f}_{\ell }^{\prime }+[x^2-\ell (\ell +1)]f_{\ell }=0$. Its regular solution is a Spherical Bessel function $j_{\ell }(x)$, up to a multiplicative constant. The second independent solution is singular as $r^{-\ell -1}$ at the origin and is excluded. HenceOne may denote $j_{\ell }$ by $f_{\ell }$ as in the source. Its leading behaviour $j_{\ell }(x)\sim x^{\ell }/(2\ell +1)!!$ ensures regularity. Primes on $j_{\ell }$ below refer to its dimensionless argument; differentiating with respect to $r$ introduces the factor $\sqrt{s}$.