Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 36B (Applications of Quantum Mechanics)

The s-wave phase shift is defined byExpanding the sine and comparing coefficients with the given expression givesso $\boxed{\tan {\delta }_0=g(k)}$.

The scattering amplitude is defined byFor a central potential its partial-wave expansion isUsing orthogonality of the Legendre polynomials in ${\sigma }_T=\int |f{|}^{2}d\Omega$ gives

Here $k\cot {\delta }_0=-K$, so the s-wave amplitude isIt has a pole at $k=iK$, corresponding to the bound-state energyMoreover,

Since $\tan {\delta }_0=\gamma /(K_0-k)$,The exact nearby stationary point satisfies $K_0-k={\gamma }^2/K_0+O({\gamma }^4/K_0^3)$, so to leading order in $\gamma /K_0$ the local maximum is at $k=K_0$. This is a scattering resonance withwhere $\Gamma$ is the full energy width of the Breit-Wigner peak.