Math Infinitum

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Intuition

Each phase shift has a plain meaning: far from the potential, the radial wave of angular momentum $\ell$ is $\sin(kr-\ell\pi/2+\delta_{\ell})$ instead of $\sin(kr-\ell\pi/2)$ , shifted by $\delta_{\ell}$ . An attractive potential pulls the wave in, a positive phase shift; a repulsive one pushes it out, a negative one. A hard sphere of radius $a$ pushes the s wave out by exactly $a$ : $\delta_{0}=-ka$ . Integrate $|f|^{2}$ over angles, and because the Legendre polynomials are orthogonal the partial waves do not interfere in the total: $\sigma=(4\pi/k^{2})\sum(2\ell+1)\sin^{2}\delta_{\ell}$ . Each term is at most $4\pi(2\ell+1)/k^{2}$ , the unitarity limit, reached when $\delta_{\ell}=\pi/2$ . However strong the potential, one partial wave cannot scatter more.

A column of marchers crossing a patch of mud comes out a step behind, and one crossing a moving walkway a step ahead; which it is tells what they crossed. The phase shift is that step, lost or gained by the wave.

The $s$ -wave radial function $u=rR$ of a free particle, $\sin kr$ , dashed, and outside a hard sphere of radius $a$ , $\sin k (r - a$ , solid, with . The wall, the vertical line, pushes the whole wave out by : a phase shift .

Phase shifts

Far from the potential the radial function of partial wave $\ell$ is shifted by $\delta_{\ell}$ , and the cross section is a sum over partial waves:

\begin{aligned}u_{\ell}(r)&\propto\sin\Bigl(kr-\frac{\ell\pi}{2}+\delta_{\ell}\Bigr)\\ \sigma&=\frac{4\pi}{k^{2}}\sum_{\ell}(2\ell+1)\sin^{2}\delta_{\ell}\end{aligned}

Properties

An attractive potential gives $\delta_{\ell}>0$ , pulling the wave in; a repulsive one gives $\delta_{\ell}<0$ .

The cross section from the phase shifts

\begin{aligned}\sigma&=\sum_{\ell}\sigma_{\ell},\qquad \sigma_{\ell}=\frac{4\pi}{k^{2}}(2\ell+1)\sin^{2}\delta_{\ell}\\ \sigma_{\ell}&\le\frac{4\pi(2\ell+1)}{k^{2}}\end{aligned}

Square the partial-wave sum and integrate over directions. The Legendre polynomials are orthogonal, so the cross terms between different partial waves vanish, and each contributes on its own. Its contribution is largest when $\sin^{2}\delta_{\ell}=1$ .

Proof steps

\sigma=\int|f|^{2}\,d\Omega=2\pi\int_{-1}^{1}|f|^{2}\,d(\cos\theta)

$f$ does not depend on $\phi$ .

|f|^{2}=\frac{1}{k^{2}}\sum_{\ell,\ell'}(2\ell+1)(2\ell'+1)\,a_{\ell}a_{\ell'}^{*}\,P_{\ell}P_{\ell'}

With $a_{\ell}=e^{i\delta_{\ell}}\sin\delta_{\ell}$ ; the Legendre polynomials are real.

\int_{-1}^{1}P_{\ell}P_{\ell'}\,dx=\frac{2}{2\ell+1}\,\delta_{\ell\ell'}

The Legendre polynomials are orthogonal.

\begin{aligned}\sigma&=\frac{2\pi}{k^{2}}\sum_{\ell}(2\ell+1)^{2}\,\frac{2\,|a_{\ell}|^{2}}{2\ell+1}\\ &=\frac{4\pi}{k^{2}}\sum_{\ell}(2\ell+1)\sin^{2}\delta_{\ell}\end{aligned}

Only $\ell=\ell'$ survives, and $|a_{\ell}|^{2}=\sin^{2}\delta_{\ell}$ .

\sigma_{\ell}\le\frac{4\pi(2\ell+1)}{k^{2}},\ \ \text{equal at}\ \delta_{\ell}=\frac{\pi}{2}

$\sin^{2}\delta_{\ell}\le1$ .

Applications

Practice

The Unitarity Limit

One partial wave can scatter at most $4\pi(2\ell+1)/k^{2}$ , reached when its phase shift is $\pi/2$ .

\sigma_{\ell}\le\frac{4\pi(2\ell+1)}{k^{2}}

Try it

What is the largest possible $s$ -wave cross section at $k=1$ fm $^{-1}$ , in fm²? Give two decimal places.