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Unformatted text preview: Phys 852, Quantum mechanics II, Spring 2008 Scattering theory 2/25/2008 Prof. Michael G. Moore, Michigan State University 1 Statement of the Problem: Scattering theory is essentially time-independent perturbation theory applied to the case of a continuous spectrum. We assume an incident particle in state | ) subject to the background Hamiltonian H , which is scattered (i.e. perturbed) by the potential V . The incident state | ) is assumed to be an eigenstate of H , with eigenvalue E , satisfying ( E H ) | ) = 0 . (1) Unless otherwise specified, the background Hamiltonian should be taken as that of a free-particle, H = P 2 2 M , (2) and the incident state taken as a plane wave ( vector r | ) = ( vector r ) = (2 ) d/ 2 e i vector k vector r , (3) where d is the number of physical degrees of freedom (i.e. the number of components in vector r and vector k ). Fur- thermore, the potential V ( vector R ) is assumed to be localized, so that lim r V ( vector r ) = 0 . (4) The goal of scattering theory is then to solve the energy-eigenvalue problem ( E H V ) | ) = 0 , (5) where E > 0 (unless otherwise specified), and | ) is the eigenstate of the full Hamiltonian H = H + V with energy E . Consistent with the boundary condition that the only incoming wave component is | ) . Since the spectrum of energy eigenvalues is continuous, we do not compute shifts to the energy levels, we just need to find the perturbed eigenstates. This can be accomplished via the Lippman-Schwinger Equation, which we will derive by first introducing the retarded background Greens function G H ( E ) = ( E H + i ) 1 . (6) The i term is needed so that G H ( E ) is non-singular, and later we will see that it enforces causality by allowing a point source to emit only out-going waves. Note that is infinitesimal, and should be taken to zero at the end of a calculation, i.e. = 0 + . This leads to the relation G H ( E )( E H ) = 1 , (7) which is clearly a useful relation if one is to formally solve the energy-eigenvalue equation via operator inversion. We now proceed to do this by first defining the scattered wave | s ) via | ) = | ) + | s ) . (8) 1 In other words, the scattered wave is the piece that when added to the incident state results in an eigenstate of the full Hamiltonian. Inserting Eq. ( ?? ) into ( ?? ) and taking ( ?? ) into account gives ( E H V ) | s ) = V | ) . (9) Operating on this from the left with G H ( E ) then gives (1 G H ( E ) V ) | s ) = G H ( E ) V | ) , (10) which can be re-expressed as | s ) = G H ( E ) V ( | ) + | s ) ) , (11) which is known as the Lippman-Schwinger equation . This equation is often written in the equivalent form | ) = | ) + G H ( E ) V | ) . (12) 2 The T-Matrix The Lippman-Schwinger Equation ( ?? ) can be formally solved for | s ) , yielding | s ) = (1 G H ( E ) V ) 1 G H ( E ) V | ) . (13) The T-matrix,...
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