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Quantum Scattering by a Rigid Sphere
Quantum Scattering by a Rigid Sphere
Consider the quantum-mechanical treatment of scattering in three dimensions by a spherically symmetrical potential centered at the origin. A standard problem is scattering by a rigid sphere, defined by the potential-energy function: for , for . For any spherically symmetrical potential, the asymptotic behavior of the wavefunction can be represented by the Faxen–Holzmark formula , where the first term represents an incoming plane wave in the positive direction () and the second term is an outgoing spherical wave from the scattering center. The scattering amplitude determines the angular distribution of the outgoing wave. The differential scattering cross section is given by =σ(θ)=f(θ). A spherical-harmonic expansion of the plane wave, =(2l+1)(kr)(cosθ), reduces the outgoing wave to a sum of partial waves, the so-called partial wave expansion. The component is known as the -wave, the component, the -wave, and so on. In practice, only a small number partial waves need be considered for a sufficiently accurate representation of the scattering, at least at low energies. The defining parameter of a partial wave is its phase shift . For scattering by a rigid sphere, . In the preceding formulas, and are spherical Bessel functions of the first and second kind, while is a Legendre polynomial.
V(r)=∞
r⩽a
V(r)=0
r>a
ψ(r,θ)≈+f(θ)
ikz
e
ikr
e
r
z
z=rcosθ
f(θ)
dσ
dΩ
2
|
ikrcosθ
e
∞
∑
l=0
l
i
j
l
P
l
l=0
s
l=1
p
δ
l
tan=(ka)/(ka)
δ
l
j
l
y
l
j
l
y
l
P
l
It is shown in all of the cited references that, in general, . From this it follows that the total cross section is given by =(2l+1)=Imf(0), a result known as the optical theorem. For rigid-sphere scattering at low energies (), the scattering cross sections are approximated by and ≈4π, four times the classical value.
f(θ)=(2l+1)(-1)(cosθ)
1
2ik
∞
∑
l=0
2i
δ
l
e
P
l
σ
T
4π
2
k
∞
∑
l=0
2
sin
δ
l
4π
k
k0
σ(θ)≈
2
a
σ
T
2
a
This Demonstration shows a pictorial representation of the spherical wavefronts in the Faxen–Holzmark formula, which you can animate. (The angular dependence of these amplitudes is not shown.) The checkbox produces a plot of the differential scattering cross section , shown in red, which changes as and are varied.
σ(θ)
k
a