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852HW10_Solutions - HOMEWORK ASSIGNMENT 10 Solutions...

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HOMEWORK ASSIGNMENT 10: Solutions PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: partial waves, scattering resonances . 1. Hard-sphere S-wave scattering: Consider S-wave scattering from a hard sphere of radius a . Make the ansatz ψ ( r,θ,φ ) = e ikr r (1 + 2 ikf 0 ( k )) e ikr r and show that it is an eigenstate of the full Hamiltonian for all r>a . Fit the value of f 0 ( k ) to satisfy 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 )? Answer: Using ψ ( r ) = u ( r ) /r , the general s-wave free-space 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 0 ( k )) e ikr the hard-wall boundary condition ψ ( a ) = 0 reduces to: e ika (1 + 2 ikf 0 ( k )) e ika = 0 . Which leads directly to (1 + 2 ikf 0 ( a )) = e i 2 ka Using the definition 1 + 2 ikf ( k ) = e i 2 δ ( k ) we see by inspection that the s-wave phase shift is δ 0 ( k ) = ka Solving for f 0 ( k ) via the relation f ( k ) = e ( k ) sin( δ ( k )) /k , we find f 0 ( k ) = e ika sin( ka ) k . A key point not to overlook in this problem, is that for the hard-sphere, the s-wave hscattering length is equal to the radius of the sphere. 1
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2. Hard-sphere P-wave scattering: For P-wave 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 0 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 phase-shift δ 1 ( k )? Show that it behaves as ( ka ) 3 . This is a general result that for small k we have δ ( k ) k 2 +1 , called ‘threshold behavior. Take the limit as k 0 and show that δ 1 ( k ) is negligible compared to δ 0 ( k ). This is an example of how higher partial waves are ‘frozen out’ at low energy. Answer: To show that the ansatz satisfies the free-particle 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 parenrightbigg parenleftbigg 1 + i kr parenrightbigg e ikr = 0 .
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