At low energy only the s wave scatters, and its phase shift vanishes in proportion to $k$: $\delta_{0}\approx-ka$. The length $a$, the scattering length, then settles everything: $f\to-a$ and $\sigma\to4\pi a^{2}$, the same at every angle and every low energy. For a hard sphere $a$ is its radius, so slow quantum particles see four times the classical cross section. For an attractive well $a$ can have either sign and any size. It grows without limit when the well is just deep enough to hold a new bound state at zero energy — a resonance at threshold — and it vanishes at special depths, where slow particles pass as if nothing were there. At higher energies a partial wave can resonate: $\delta_{\ell}$ climbs through $\pi/2$ within an energy range $\Gamma$ about $E^{*}$, the cross section peaks at the unitarity limit with the Breit–Wigner shape, and the particle lingers in the potential for a time $\hbar/\Gamma$, a quasi-bound state.

A swing pushed in time with its own rhythm swings high and keeps swinging after the pushes stop; pushed out of time it barely moves. A resonance is the potential’s rhythm: particles at just the right energy are held a while before they leave.

The scattering length of a square well of radius $R$, in units of $R$, against its strength $K_{0}R=\sqrt{2mV_{0}}\,R/\hbar$. It diverges at , dashed, where a bound state appears at zero energy, and it vanishes at , where slow particles pass through the well unscattered.

As $k\to0$ the $s$-wave phase shift defines the scattering length $a$; near a resonance of partial wave $\ell$ at the energy $E_{*}$, of width :

• In general $\delta_{\ell}\propto k^{2\ell+1}$ as $k\to0$: the higher partial waves die out fast at low energy.

Solve for the s wave inside and outside and match the logarithmic derivative at the edge. At low energy the outside wave is a straight line, $u\propto r-a$, whose zero is the scattering length; the inside wave is a sine with the well’s wavenumber. Matching fixes $a$.

Inside the well, with $u=rR$ vanishing at the centre.

With $\delta_{0}\approx-ka$, $\sin(kr-ka)\approx k(r-a)$.

The logarithmic derivative $u'/u$ is continuous at $r=R$.

Solve for $a$.

A new bound state at zero energy, and a well that slow particles do not see.

At low energy $f$ tends to $-a$, a length called the scattering length, and the cross section to $4\pi a^{2}$, the same in every direction.

As $k\to0$, $\delta_{\ell}$ vanishes as $k^{2\ell+1}$: the centrifugal barrier keeps slow particles with away from the potential.

A hard sphere of radius $R$ has scattering length $R$, so slow quantum particles see $4\pi R^{2}$, four times the classical cross section.

The scattering length of a well diverges when the well is just deep enough to hold a new bound state at zero energy.

Great work! You now know how to:

• define the scattering length and the low-energy cross section
• derive the scattering length of a square well
• describe a resonance by its energy and width