Existence of Schrodinger Evolution with Absorbing Boundary Condition

• URL: http://arxiv.org/abs/1912.12057v2
• Date: Fri, 29 Jul 2022 11:55:17 GMT
• Title: Existence of Schrodinger Evolution with Absorbing Boundary Condition
• Authors: Stefan Teufel, Roderich Tumulka
• Abstract summary: Consider a non-relativistic quantum particle with wave function inside a region $Omegasubset mathbbR3$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface. A particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle's wave function $psi$ expressed by a Schrodinger equation in $Omega$ together with an "absorbing" boundary condition on $partial Omega$ first considered by Werner in
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Consider a non-relativistic quantum particle with wave function inside a region $\Omega\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial \Omega$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle's wave function $\psi$ expressed by a Schrodinger equation in $\Omega$ together with an "absorbing" boundary condition on $\partial \Omega$ first considered by Werner in 1987, viz., $\partial \psi/\partial n=i\kappa\psi$ with $\kappa>0$ and $\partial/\partial n$ the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of $\psi$; we point out here how the Hille-Yosida theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the $N$-particle version of the problem is well defined. Finally, we also prove analogous results for the Dirac equation.

Related papers

• Hamiltonians for Quantum Systems with Contact Interactions [49.1574468325115]
  We show that in the limit one obtains the one-body Hamiltonian for the light particle subject to $N$ (non-local) point interactions placed at fixed positions. We will verify that such non-local point interactions do not exhibit the ultraviolet pathologies that are present in the case of standard local point interactions.
  arXiv  Detail & Related papers  (2024-07-09T14:04:11Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• Small-time controllability for the nonlinear Schr\"odinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields [55.2480439325792]
  We address the small-time controllability problem for a nonlinear Schr"odinger equation (NLS) on $mathbbRN$ in the presence of magnetic and electric external fields. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals.
  arXiv  Detail & Related papers  (2023-07-28T21:30:44Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• Non-perturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field [0.0]
  We prove existence and uniqueness of classical solutions of the Schr"odinger equation for general initial conditions. We show that the solution approaches in the large $t$ limit a periodic state that satisfies an infinite set of equations.
  arXiv  Detail & Related papers  (2022-09-15T19:14:53Z)
• On parametric resonance in the laser action [91.3755431537592]
  We consider the selfconsistent semiclassical Maxwell--Schr"odinger system for the solid state laser. We introduce the corresponding Poincar'e map $P$ and consider the differential $DP(Y0)$ at suitable stationary state $Y0$.
  arXiv  Detail & Related papers  (2022-08-22T09:43:57Z)
• An Introduction to Scattering Theory [0.0]
  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)
  arXiv  Detail & Related papers  (2022-04-08T11:41:24Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• The Principle of equal Probabilities of Quantum States [0.0]
  Boltzmann law $P(epsilon) = frac1langle epsilon rangle-fracepsilonlangle epsilon rangle ; ; ;;;; 0leq epsilon +infty$ where $langle epsilon rangle = E/N$. kappa quanta, is given by $p(kappa)=fracdisplaystyle binomN+s
  arXiv  Detail & Related papers  (2021-11-17T17:23:46Z)
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots [0.0]
  $mathttd$-dimensional hyperspherical quantum dot with either Dirichlet or Neumann boundary conditions (BCs) This paves the way to an efficient computation in either space of Shannon, R'enyi and Tsallis entropies, Onicescu energies and Fisher informations.
  arXiv  Detail & Related papers  (2021-03-24T11:08:31Z)
• Energy-Time Uncertainty Relation for Absorbing Boundaries [0.0]
  We prove the uncertainty relation $sigma_T, sigma_E geq hbar/2$ between the time $T$ of detection of a quantum particle on the surface.
  arXiv  Detail & Related papers  (2020-05-29T12:04:57Z)

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.