Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 35E / a

For real $k$, conservation of the radial probability current, equivalently constancy of the radial Wronskian, equates the incoming and outgoing fluxes and givesFor a real potential, the regular radial solution depends on $k$ through $k^2$. Comparing its asymptotic form at $k$ and $-k$ therefore givesThe first identity places $S_l(k)$ on the unit circle, sofor a real phase shift. Choosing the phase continuously, the second identity givesThese are the unitarity and reflection identities for a partial-wave S-matrix.