Scattering_II - More Scattering: the Partial Wave Expansion...

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More Scattering: the Partial Wave Expansion Michael Fowler 1/17/08 Plane Waves and Partial Waves We are considering the solution to Schrödinger’s equation for scattering of an incoming plane wave in the z -direction by a potential localized in a region near the origin, so that the total wave function beyond the range of the potential has the form () ( ) cos ,, , . ikr ikr e re f r θ ψ θϕ =+ The overall normalization is of no concern, we are only interested in the fraction of the ingoing wave that is scattered. Clearly the outgoing current generated by scattering into a solid angle at angle θ , φ is d Ω 2 , f d Ω multiplied by a velocity factor that also appears in the incoming wave. Many potentials in nature are spherically symmetric, or nearly so, and from a theorist’s point of view it would be nice if the experimentalists could exploit this symmetry by arranging to send in spherical waves corresponding to different angular momenta rather than breaking the symmetry by choosing a particular direction. Unfortunately, this is difficult to arrange, and we must be satisfied with the remaining azimuthal symmetry of rotations about the ingoing beam direction. In fact, though, a full analysis of the outgoing scattered waves from an ingoing plane wave yields the same information as would spherical wave scattering. This is because a plane wave can actually be written as a sum over spherical waves : .c o s (2 1) ( ) (cos ) ik r ikr l ll l ee i l j k r P == + G G Visualizing this plane wave flowing past the origin, it is clear that in spherical terms the plane wave contains both incoming and outgoing spherical waves. As we shall discuss in more detail in the next few pages, the real function ( ) l j kr is a standing wave, made up of incoming and outgoing waves of equal amplitude. We are, obviously, interested only in the outgoing spherical waves that originate by scattering from the potential , so we must be careful not to confuse the pre-existing outgoing wave components of the plane wave with the new outgoing waves generated by the potential. The radial functions ( ) l j kr appearing in the above expansion of a plane wave in its spherical components are the spherical Bessel functions , discussed below. The azimuthal rotational symmetry of plane wave + spherical potential around the direction of the ingoing wave ensures that the angular dependence of the wave function is just (cos ) l P , not ( ) , lm Y . The coefficient
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2 ( ) 21 l il + is derived in Landau and Lifshitz, §34, by comparing the coefficient of () cos n kr θ on the two sides of the equation: as we shall see below, ( does not appear in ) n kr ( ) l j kr for l greater than n , and cos n does not appear in (cos ) l P for l less than n , so the combination ( cos n kr ) only occurs once—in the n th term, and the coefficients on both sides of the equation can be matched. (To get the coefficient right, we must of course specify the normalizations for the Bessel function—see below—and the Legendre polynomial.) Mathematical Interval: The Spherical Bessel and Neumann Functions The plane wave
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This note was uploaded on 12/07/2011 for the course PHYSICS 751 taught by Professor Michaelfowler during the Fall '07 term at UVA.

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Scattering_II - More Scattering: the Partial Wave Expansion...

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