An Introduction to Scattering Theory
- URL: http://arxiv.org/abs/2204.03651v1
- Date: Fri, 8 Apr 2022 11:41:24 GMT
- Title: An Introduction to Scattering Theory
- Authors: Milan \v{S}indelka
- Abstract summary: Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain.
Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain.
Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism)
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
Related papers
- The Dirac Delta as a Singular Potential for the 2D Schrodinger Equation [0.0]
In the framework of distributionally generalized quantum theory, the object $Hpsi$ is defined as a distribution.
The significance is a mathematically rigorous method, which does not rely upon renormalization or regularization of any kind.
The distributional interpretation resolves the need to evaluate a wave function at a point where it fails to be defined.
arXiv Detail & Related papers (2023-12-23T00:43:06Z) - Quantum tomography of helicity states for general scattering processes [55.2480439325792]
Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics.
We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
arXiv Detail & Related papers (2023-10-16T21:23:42Z) - Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance.
The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
arXiv Detail & Related papers (2023-05-22T11:00:25Z) - One-dimensional pseudoharmonic oscillator: classical remarks and
quantum-information theory [0.0]
Motion of a potential that is a combination of positive quadratic and inverse quadratic functions of the position is considered.
The dependence on the particle energy and the factor $mathfraka$ describing a relative strength of its constituents is described.
arXiv Detail & Related papers (2023-04-13T11:50:51Z) - Existence of the transfer matrix for a class of nonlocal potentials in
two dimensions [0.0]
Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction.
In potential scattering defined by the Schr"odinger equation, $(-nabla2+v)psi=k2psi$ for a local potential $v$, they arise in dimensions greater than one and are present regardless of the details of $v$.
arXiv Detail & Related papers (2022-07-20T17:34:05Z) - $\PT$ Symmetry and Renormalisation in Quantum Field Theory [62.997667081978825]
Quantum systems governed by non-Hermitian Hamiltonians with $PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution.
We show how $PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework.
arXiv Detail & Related papers (2021-03-27T09:46:36Z) - Universal scattering with general dispersion relations [0.15749416770494704]
We show that when there are no bright zero-energy eigenstates, the $S$-matrix evaluated at an energy $Eto 0$ converges to a universal limit.
We extend these results to general integer dimensions $D geq 1$, dispersion relations $epsilon(boldsymbolk) = |boldsymbolk|a$ for a $D$-dimensional momentum vector.
arXiv Detail & Related papers (2021-03-17T18:00:03Z) - Topological Quantum Gravity of the Ricci Flow [62.997667081978825]
We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow.
First, we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric.
We extend the primitive theory by gauging foliation-preserving spacetime symmetries.
arXiv Detail & Related papers (2020-10-29T06:15:30Z) - Quantum dynamics and relaxation in comb turbulent diffusion [91.3755431537592]
Continuous time quantum walks in the form of quantum counterparts of turbulent diffusion in comb geometry are considered.
Operators of the form $hatcal H=hatA+ihatB$ are described.
Rigorous analytical analysis is performed for both wave and Green's functions.
arXiv Detail & Related papers (2020-10-13T15:50:49Z) - Sub-bosonic (deformed) ladder operators [62.997667081978825]
We present a class of deformed creation and annihilation operators that originates from a rigorous notion of fuzziness.
This leads to deformed, sub-bosonic commutation relations inducing a simple algebraic structure with modified eigenenergies and Fock states.
In addition, we investigate possible consequences of the introduced formalism in quantum field theories, as for instance, deviations from linearity in the dispersion relation for free quasibosons.
arXiv Detail & Related papers (2020-09-10T20:53:58Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.