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

A scattering phase shift measures the change of a partial wave's outgoing phase relative to the free radial solution. For a real central potential and elastic scattering, probability conservation gives a unit-modulus Partial-wave S-matrix $S_l=e^{2i{\delta }_l}$. Indeedso the incoming amplitude is unchanged and the outgoing one acquires $S_l$.

Expanding an incident plane wave in Legendre polynomials and comparing the outgoing spherical wave givesUse $d\sigma /d\Omega =|f{|}^2$ and $\int d\Omega P_l{P}_{l^{\prime }}=4\pi {\delta }_{l{l}^{\prime }}/(2l+1)$ to remove cross terms:This is the elastic total scattering cross-section for the one-particle central-potential problem, with no identical-particle symmetrization factor.

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.

At a finite potential step, both $u$ and $u^{\prime }$ are continuous:Taking the logarithmic derivative eliminates the amplitudes:At a zero of a denominator, the original two matching equations provide the continuous limiting condition.

Write $\kappa =\sqrt{2m{V}_0}/\hslash$ and $q=\kappa a$. At low energy, the $S$ wave normally dominates. Under the additional stated near-resonance assumption $ka\ll |{\delta }_0|$, neglect $ka$ inside the cotangent of the matching condition and set $k^{\prime }\simeq \kappa$:Substitution in ${\sigma }_0=4\pi {\sin }^2{\delta }_0/k^2$ givesThis is a near-resonance square-well approximation, not a uniform formula for every well radius. A phase shift is defined modulo $\pi$; its representative must be chosen consistently when applying the comparison with $ka$.

For small fixed positive $E$, the stated approximate curve has maxima atIts formal minima are zero at $\kappa a=n\pi$. The peak height is the $S$-wave unitarity limit, and the peaks narrow as $E/V_0$ decreases.

To identify the bound-state threshold, let the bound-state energy be $-\epsilon$ with $\epsilon >0$. The reduced radial wave is proportional to $\sin (Kr)$ inside and $e^{-\chi r}$ outside, with $K=\sqrt{2m(V_0-\epsilon )}/\hslash$ and $\chi =\sqrt{2m\epsilon }/\hslash$. Matching gives $K\cot (Ka)=-\chi$. As a new bound state emerges at $\epsilon \downarrow 0$,Thus the limiting peak locations coincide with the appearance of successive $S$-wave bound states. At the exact threshold the zero-energy radial solution is not square-integrable; a normalizable bound state appears just beyond it. Finite-energy maxima shift slightly from the zero-energy positions.

The formal minima require care because the assumption used in part d fails there. Retaining $ka$ in the exact matching condition gives the spherical square-well scattering lengthaway from a threshold. At $\kappa a=n\pi$, $a_s=a$, so the true low-energy S-wave scattering contribution is $4\pi a^2$, not zero. Genuine leading-order zeros of that contribution instead satisfy $\tan q=q$, for example $q\simeq 4.493409$ and $7.725252$. The original sketch compares the exact finite-energy $S$ wave with the requested approximation, making this nonuniformity visible. Dominance by the $S$ wave must be checked separately near a higher-partial-wave threshold. The scattering-length divergence and its bound-state interpretation are also derived in institutional lecture notes.^[ref]