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852HW8 - HOMEWORK ASSIGNMENT 8 PHYS852 Quantum Mechanics II...

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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 + ) 1 For case H = H 0 + V , there is also a ‘background Green’s function’: G H 0 ( E ) = ( E H 0 + ) 1 What is the relationship between G H ( E ) and G H 0 ( E )? To answer this, start from the Schr¨odinger equation ( E H 0 V ) | ψ s ) = V | ψ 0 ) 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 0 T H 0 ,V ( E ) | ψ 0 ) to show that G H ( E ) V = G H 0 ( E ) T H 0 ,V ( E ) . Insert the general solution T H 0 ,V ( E ) = (1 VG H 0 ( E )) 1 V and derive an expression for G H ( E ) in terms of G H 0 ( E ) and V . Now use the definition of the operator inverse to derive the expression (1 VG H 0 ( E )) 1 = 1 + T H 0 ,V ( E ) G H 0 ( E ) Plug this into your previous expression for G H ( E ) to show finally, that G H ( E ) = G H 0 ( E ) + G H 0 ( E ) T H 0 ,V ( E ) G H 0 ( E ) 1
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2. Finding new bound-states : Consider a system described by
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