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Unformatted text preview: HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Green’s functions, T-matrix . 1. The full Green’s function : A system with hamiltonian H has a Green’s function defined by G H ( E ) = ( E − H + iǫ ) − 1 For case H = H + V , there is also a ‘background Green’s function’: G H ( E ) = ( E − H + iǫ ) − 1 What is the relationship between G H ( E ) and G H ( E )? To answer this, start from the Schr¨odinger equation ( E − H − V ) | ψ s ) = V | ψ ) and operate on both sides with the full Green’s function G H ( E ). Then compare your result with the T-matrix definition | ψ s ) = G H T H ,V ( E ) | ψ ) to show that G H ( E ) V = G H ( E ) T H ,V ( E ) . Insert the general solution T H ,V ( E ) = (1 − V G H ( E )) − 1 V and derive an expression for G H ( E ) in terms of G H ( E ) and V . Now use the definition of the operator inverse to derive the expression (1 − V G H ( E )) − 1 = 1 + T H ,V ( E ) G H ( E ) Plug this into your previous expression for G H ( E ) to show finally, that...
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