# Quantum Scattering by a Rigid Sphere

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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 . A spherical-harmonic expansion of the plane wave, , 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.

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

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Reference: This topic is treated in most quantum-mechanics textbooks, notably those of Schiff, Merzbacher, and Landau–Lifshitz.

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