An Introduction to Scattering Theory

The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator $\hat{S}$). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as $| ψ_{E(\pm 1)}^\pm \rangle$), to discuss fundamental properties of $| ψ_{E(\pm 1)}^\pm \rangle$, and subsequently to construct $\hat{S}$ and $\hat{T}$ operators in terms of $| ψ_{E(\pm 1)}^\pm \rangle$. Physical contents of the $\hat{S}$ and $\hat{T}$ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.

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