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852_2010hw12_Solutions

852_2010hw12_Solutions - PHYS852 Quantum Mechanics II...

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Unformatted text preview: 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 ( k )) e ikr r Then, find the value of f ( k ) that satisfies the boundary condition ψ ( a, θ, φ ) = 0. What is the partial amplitude f ( k )? What is the s-wave phase-shift δ ( k )? Satisfying the required boundary condition at r = a requires 0 = e − ika − (1 + 2 ikf ( k )) e ika , (1) which gives the s-wave partial amplitude as f ( k ) = − e − ika sin( ka ) k (2) The phase-shift is related to the partial amplitude via Eqs. (137) or (138) in the lecture notes, which give δ ( k ) = − ka (3) From Eq. (137) in the notes, it then follows that the scattering length is a . Thus we can interpret the scattering length of a particular scatterer as the radius of the hard-sphere whose scattering amplitude matches that of the scatterer in the low energy ( k → 0) limit. 1 2. 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 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 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....
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852_2010hw12_Solutions - PHYS852 Quantum Mechanics II...

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