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Unformatted text preview: HOMEWORK ASSIGNMENT 10: Solutions PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: partial waves, scattering resonances . 1. Hardsphere Swave scattering: Consider Swave scattering from a hard sphere of radius a . 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 > a . Fit the value of f ( k ) to satisfy the boundary condition ψ ( a,θ,φ ) = 0. What is the partial amplitude f ( k )? What is the swave phaseshift δ ( k )? Answer: Using ψ ( r ) = u ( r ) /r , the general swave freespace eigenstate is u ( r ) = Ae − ikr + Be ikr The boundary condition at infinity requires we choose A and B so that u ( r ) = e − ikr − (1 + 2 if ( k )) e ikr the hardwall boundary condition ψ ( a ) = 0 reduces to: e − ika − (1 + 2 ikf ( k )) e ika = 0 . Which leads directly to (1 + 2 ikf ( a )) = e − i 2 ka Using the definition 1 + 2 ikf ℓ ( k ) = e i 2 δ ℓ ( k ) we see by inspection that the swave phase shift is δ ( k ) = − ka Solving for f ( k ) via the relation f ℓ ( k ) = e − iδ ℓ ( k ) sin( δ ℓ ( k )) /k , we find f ( k ) = − e − ika sin( ka ) k . A key point not to overlook in this problem, is that for the hardsphere, the swave hscattering length is equal to the radius of the sphere. 1 2. Hardsphere Pwave scattering: For Pwave scattering from a hard sphere of radius a , 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 > a . Again solve for the partial amplitude f 1 ( k ) by imposing the boundary condition ψ ( a,θ ) = 0. What is the phaseshift δ 1 ( k )? Show that it behaves as ( ka ) 3 . This is a general result that for small k we have δ ℓ ( k ) ∝ k 2 ℓ +1 , called ‘threshold behavior. Take the limit as k → 0 and show that δ 1 ( k ) is negligible compared to δ ( k ). This is an example of how higher partial waves are ‘frozen out’ at low energy. Answer: To show that the ansatz satisfies the freeparticle Hamiltonian for r > R we let ψ ( kr ) = u ( kr ) /kr . Then we have u ( kr ) = parenleftbigg 1 − i kr parenrightbigg e − ikr − +(1 + 2 ikf 1 ( k )) parenleftbigg 1 + i kr parenrightbigg e ikr To show that this is a solution, we only need to show that parenleftbigg E + planckover2pi1 2 2 M d 2 dr 2 − 1 Mr 2 parenrightbiggparenleftbigg 1 + i kr parenrightbigg...
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This note was uploaded on 10/25/2010 for the course PHYSICS PHYS 851 taught by Professor Michaelmoore during the Fall '08 term at Michigan State University.
 Fall '08
 MichaelMoore
 mechanics, Work

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