Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 6 / b / Solution

For $l=0$, put $u(r)=r{R}_0(r)$. The radial Schrödinger equation reduces to $u^{\prime \prime }+k^{\prime 2}u=0$ inside and $u^{\prime \prime }+k^{2}u=0$ outside, where $k^{\prime 2}=2m(E+V_0)/{\hslash }^2$ and $k^2=2mE/{\hslash }^2$. Regularity at the origin requires $u(0)=0$, excluding the inner cosine. A real outer sine-cosine combination can be written with an amplitude and a phase shift:The inner expression has the finite limiting value $A{k}^{\prime }$ at the origin. An overall complex normalization can carry the asymptotic phase convention of part a.