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Physics LibreTexts

14.4: Example of S-wave scattering - Attractive square well potential

( \newcommand{\kernel}{\mathrm{null}\,}\)

An example where we can solve for the phase shift is the 3D-square well potential:

(V(r<R)=V0;V(r>R)=0).

For the l=0 case the radial equation with U0=R0r is

d2u0(r)dr2+2μ2[EV(r)]u0(r)=0

The solutions to this are familiar from the 1D square well. If we write

K0=2μ[E+V0]/;K=2μE/

then for r<R,u(r)=AsinK0r+BcosK0r.

and for r>R,u(r)=CsinKr+DcosKr. which can easily be written in a different form to show the appropriate phase shift δ0:u(r)=Fsin(Kr+δ0) where (C=Fcosδ0;D=Fsinδ0)

As with the 1D square well, the boundary conditions are that u and dudr are continuous at R, which lead to:

KtanK0R=K0tan(KR+δ0) or δ0=tan1(KK0tanK0R)KR

In the low energy case KR1, we obtain maximum scattering (sin2δ01) when K0R=(n+12)π, when the scattering cross section is σ=4π/K2. This is an example of s-wave resonance.

In the same slow particle limit KK0, and assuming that tanK0R is not very large: δ0sinδ0.

σ4πR2(tanK0RK0R1)2

This correctly predicts that when tanK0R=K0R the scattering cross section will be zero.

There are a few features of the square-well which also apply in more general cases. Assuming K0 is basically a measure of the potential depth.

  • For weak coupling K0R1,δ0(K)0 as K0
  • When K0R approaches π/2 the potential is almost able to bind an s-wave bound state. Now the phase shift δ0(K)π/2 and the cross section diverges like K2 as K0. This is known as zero energy resonance.
  • If E is high enough that δl=(n+12)π for l0 the scattering cross section can become especially high due to another angular momentum component - p-wave resonance for l=1, d-wave resonance for l=2 etc. In these cases the eigenfunction becomes large near to the potential. The potential is said to have virtual states at the resonance energies.
  • Levinson’s Theorem states that limk0δl(k)=nlπ

where nl is the number of bound states with angular momentum l.

  • Whenever δ0(K)=nπ, for s-wave scattering, σ=0. Thus for certain energies of the incoming particle, the scattering is extremely small. This condition can only be consistent with the condition for s-wave scattering (KR1) if the potential is attractive (V0<0).
  • δ0(K) tends to decrease with increasing K. This can be understood physically as the faster particles having less time to interact and thus experiencing smaller phase shifts. As K,δl(K)0 because the potential is now weak relative to the particle energy. Of course σ(K) decreases even more quickly because of the K2 term.

This page titled 14.4: Example of S-wave scattering - Attractive square well potential is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.

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