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+2μℏ2[E−V(r)]u0(r)=0
The solutions to this are familiar from the 1D square well. If we write
K0=√2μ[E+V0]/ℏ;K=√2μE/ℏ
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=tan−1(KK0tanK0R)−KR
In the low energy case KR≪1, we obtain maximum scattering (sin2δ0→1) 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 K≪K0, and assuming that tanK0R is not very large: δ0≈sinδ0.
σ≈4πR2(tanK0RK0R−1)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 K0R≪1,δ0(K)→0 as K→0
- 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 K−2 as K→0. This is known as zero energy resonance.
- If E is high enough that δl=(n+12)π for l≠0 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 limk→0δl(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 (KR≪1) 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 K−2 term.