# answer%20HW8 - HW 8 Solutions Problem 1: (i)We note that...

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HW 8 Solutions Problem 1: (i)We note that the dependence on θ for a p-wave is P l ( θ ) = cos( θ ). Now, let R l ( r ) = χ ( r ) /r , then the radial equation tells us that d 2 χ dr 2 + ± k 2 - 1(1 + 1) r 2 ² χ = 0 (1) In our case, note that χ ( r ) = ³ 1 + i kr ´ e ikr . Substituting this into equation (1) directly shows that (1) is satisﬁed. (ii)Note that the term in A is the spherically symmetric s-wave and the term in B is the p-wave. As r → ∞ , φ e ikz + f ( θ ) r e ikr (2) Equating the terms in (2) and φ ( r, θ ) = e ikz + A r e ikr + B r ± 1 + i kr ² e ikr cos θ (3) yields f ( θ ) = A + B cos θ (4) (iii)First, note that since the scattering object in a hard sphere, the wavefunction vanishes on the surface r = R . Thus, e ikR cos θ + A R e ikR + B R ± 1 + i kR ² e ikR cos θ = 0 (5) Expanding e ikR cos θ in powers of kR , we have e ikR cos θ = 1 + ikR cos θ - 1 2 ( kR ) 2 cos 2 θ + O ( k 3 R 3 ) (6) The terms on the LHS of (5) must sum to zero. Note that since we are excluding contributions

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## This note was uploaded on 11/21/2010 for the course PHYSICS 137B taught by Professor Moore during the Spring '07 term at Berkeley.

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answer%20HW8 - HW 8 Solutions Problem 1: (i)We note that...

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