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852scattering

# 852scattering - Phys 852 Quantum mechanics II Spring 2008...

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Phys 852, Quantum mechanics II, Spring 2008 Introduction to Scattering Theory Statement of the problem: Scattering theory is essentially time-independent perturbation theory applied to the case of a continuous spectrum. That means that we know there is an eigenstate of the full Hamiltonian for every possible energy, E . Thus the job of finding the full eigenvalues, which was a major part of TIPR, is not necessary here. In scattering theory, we just pick any E , and then try to find the ‘perturbed’ eigenstate, | ψ ( E ) ) . On the other hand, remember that there are usually multiple degenerate eigenstates for any given energy. So the question becomes; which of the presumably infinitely many degenerate full-eigenstates are we trying to compute? The answer comes from causality; we want to be able to completely specify the probability current amplitude coming in from vector r = , and then we want the theory to give us the corresponding outgoing current amplitude. The way we do this is to pick an ‘unperturbed’ eigenstate which has the desired incoming current amplitude (we don’t need to worry what the outgoing current amplitude of the unperturbed state is). The second step is to make sure that our perturbation theory generates no additional incoming currents, which we accomplish by putting this condition in by hand, under the mantra of ‘causality’. As we will see, this means that the resulting ‘full eigenstate’ will have the desired incoming current amplitude. Now if you go back to what you know, you will recall that ‘solving’ a partial differential equation requires first specifying the desired boundary conditions, which is exactly what the standard scattering theory formalism is designed to do. Typically, the scattering formalism is described in the following way: an incident particle in state | ψ 0 ) is scattered by the potential V , resulting in a scattered state | ψ s ) . The incident state | ψ 0 ) is assumed to be an eigenstate of the ‘background’ hamiltonian H 0 , with eigenvalue E . This is expressed mathematically as ( E H 0 ) | ψ 0 ) = 0 . (1) Unless otherwise specified, the background Hamiltonian should be taken as that of a free-particle, H 0 = P 2 2 M , (2) and the incident state taken as a plane wave ( vector r | ψ 0 ) = ψ 0 ( vector r ) = e i vector k · vector r . (3) As with one-dimensional scattering, we do not need to worry about the normalization of the incident state. Furthermore, 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 full energy-eigenstate problem ( E H 0 V ) | ψ ) = 0 , (5) where E > 0 (unless otherwise specified), and | ψ ) is the eigenstate of the full Hamiltonian H = H 0 + V with energy E . It should be clear that there is a different | ψ 0 ) and correspondingly, a different | ψ ) for each energy E , even though our notation does not indicate this explicitly.

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