hw8 - HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II...

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Unformatted text preview: HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II, Spring 2008 New topics covered: Scattering amplitude, cross-section, partial wave expansion . 1. Spherical Bessel functions: The spherical Bessel function j ℓ ( ρ ) is defined as j ℓ ( ρ ) = (- 1) ℓ ρ ℓ parenleftbigg 1 ρ d dρ parenrightbigg ℓ sin ρ ρ . Show that this state satisfies the radial wave equation. 2. Hard-sphere S-wave scattering: Consider S-wave scattering from a hard sphere of radius R . Make the ansatz ψ ( r,θ,φ ) = e- ikr r- (1 + 2 ikf ( k )) e ikr r and show that it is an eigenstate of the full Hamiltonian for all r > R . Fit the value of f ( k ) to satisfy the boundary condition ψ ( R,θ,φ ) = 0. What is the partial amplitde f ( k )? What is the s-wave phase-shift δ ( k )? 3. Hard-sphere P-wave scattering: For P-wave scattering from a hard sphere of radius R , make the ansatz ψ ( r,θ ) = bracketleftbiggparenleftbigg 1 kr- i ( kr ) 2 parenrightbigg e- ikr + (1 + 2 ikf 1 ( k )] parenleftbigg 1 kr + i ( kr ) 2 parenrightbigg e ikr bracketrightbigg Y 1 ( θ ) , and show that it is an eigenstate of the full Hamiltonian for r > R . Again solve for the partial amplitude f 1 ( k ) by imposing the boundary condition ψ ( R,θ ) = 0. What is the phase-shift δ 1 ( k )?...
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This note was uploaded on 11/26/2010 for the course PHYSICS PHYS 852 taught by Professor Michaelmoore during the Spring '10 term at Michigan State University.

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hw8 - HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II...

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