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852_2010hw11

# 852_2010hw11 - PHYS852 Quantum Mechanics II Spring 2010...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 11 Topics covered: Scattering amplitude, differential cross-section, scattering probabilities. 1. Using only the definition, G 0 = ( E H 0 + ) 1 , show that the free-space Green’s function is the solution to bracketleftbigg E + planckover2pi1 2 2 M 2 vector r bracketrightbigg G 0 ( vector r,vector r ) = δ 3 ( vector r vector r ) . (1) The purpose of this problem is just to establish the equivalence between our operator-based approach, and the standard Green’s function formalism encountered, e.g., in classical EM. 2. If we define the operator F via f ( vector k , vector k ) = ( vector k | F | vector k ) , then it follows that F = (2 π ) 2 M planckover2pi1 2 T , where T is the T-matrix operator. In principle, one would like to deduce the form of the potential V from scattering data. First, derive an expression for the operator V in terms of the operators G 0 and T only.

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