852_2010hw12

# 852_2010hw12 - PHYS852 Quantum Mechanics II Spring 2010...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 12 Topics covered: Partial waves. 1. Consider S-wave scattering from a hard sphere of radius a . First, make the standard s-wave scattering ansatz: ψ ( r,θ,φ ) = e - ikr r - (1 + 2 ikf 0 ( k )) e ikr r Then, ﬁnd the value of f 0 ( k ) that satisﬁes the boundary condition ψ ( a,θ,φ ) = 0. What is the partial amplitude f 0 ( k )? What is the s-wave phase-shift δ 0 ( k )? 2. For P-wave scattering from a hard sphere of radius a , make the ansatz ψ ( r,θ ) = ±² 1 kr - i ( kr ) 2 ³ e - ikr + (1 + 2 ikf 1 ( k )) ² 1 kr + i ( kr ) 2 ³ e ikr ´ Y 0 1 ( θ. Verify that this is an eigenstate of the full Hamiltonian for r > a by showing that it is a linear superposition of two spherical Bessel functions of the third-kind. Again solve for the partial amplitude, f 1 ( k ), by imposing the boundary condition ψ ( a,θ,φ ) = 0. What is the phase-shift δ 1 ( k )? Show that it scales as ( ka ) 3 in the limit k 0. This is a general result that for small
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