In this article we provide a pedagogical introduction to scattering theory in one space dimension. This is an elegant topic that is mathematically simple and physically transparent. We shall apply the Schrodinger equation to a generic system to identify the Bogoliubov coefficients. Furthermore, we shall then derive a number of significant relationships between reflection and transmission amplitudes.
In this article we provide a pedagogical introduction to scattering theory in one space dimension. This is an elegant topic that is mathematically simple and physically transparent. We shall apply the Schrodinger equation to a generic system to identify the Bogoliubov coefficients. Furthermore, we shall then derive a number of significant relationships between reflection and transmission amplitudes.
Synopsis: At this point we are equipped to solve some simple but fundamental one-dimensional problems with the Schrödinger equation. After setting the stage with an introduction, four quantum mechanical problems are specified and solved. Along the way, we encounter important concepts such as parity in the case of the infinite well and harmonic oscillator problems, transmission and reflection coefficients in the case of the finite barrier, and energy bands and gaps in a model of conduction in solids. Raising and lowering ladder operators are introduced in the harmonic oscillator and square well cases. I conclude with some qualitative comments regarding expected types of solutions in one-dimensional problems.
Synopsis: At this point we are equipped to solve some simple but fundamental one-dimensional problems with the Schrödinger equation. After setting the stage with an introduction, four quantum mechanical problems are specified and solved. Along the way, we encounter important concepts such as parity in the case of the infinite well and harmonic oscillator problems, transmission and reflection coefficients in the case of the finite barrier, and energy bands and gaps in a model of conduction in solids. Raising and lowering ladder operators are introduced in the harmonic oscillator and square well cases. I conclude with some qualitative comments regarding expected types of solutions in one-dimensional problems.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multidelta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and the Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schrodinger equation for an effective nonunitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
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 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 ∣ψE(±1)±⟩), to discuss fundamental properties of ∣ψE(±1)±⟩, and subsequently to construct S^ and T^ operators in terms of ∣ψE(±1)±⟩. Physical contents of the S^ and 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.
Synopsis: Students are introduced to the ideas of particle scattering in quantum mechanics in a one-dimensional context. e integral Schrödinger equation is deduced using a Wronskian technique, and reflection and transmission amplitudes are defined. Several simple scattering problems are solved, and the Born series for the energy-wave function is explained. A three-dimensional spherical basis is introduced, which allows the integral Schrödinger equation in this context to be formulated. e relation between plane waves and spherical waves is found from a solution to the unit-source Helmholtz equation, again using the Wronskian technique. Partial waves, phase shifts, and cross sections are defined, and finite range scattering is considered. e chapter concludes with a consideration of identical particle scattering and an application to the protonproton case, where a dramatic manifestation of the underlying substructure emerges at high energies.
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.