An Introduction to Scattering Theory

TL;DR

This paper provides a comprehensive, accessible introduction to quantum scattering theory in one dimension, covering time-dependent and time-independent approaches, and exploring resonance phenomena with practical examples.

Contribution

It offers a structured, self-contained presentation of scattering theory, including the Lippmann-Schwinger equation, operators, and resonance analysis with nonhermitian formalism.

Findings

Explicit formulas for transmission and reflection probabilities.

Numerical example illustrating scattering resonances.

Application of Siegert pseudostate formalism to resonance phenomena.

Abstract

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 $| \psi_{E(\pm 1)}^\pm \rangle$), to discuss fundamental properties of $| \psi_{E(\pm 1)}^\pm \rangle$, and subsequently to construct $\hat{S}$ and $\hat{T}$ operators in terms of $| \psi_{E(\pm 1)}^\pm \rangle$. 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.