Variational solution of the Schrödinger equation using plane waves in adaptive coordinates: The radial case

2010 ◽  
Vol 132 (2) ◽  
pp. 024110 ◽  
Author(s):  
José M. Pérez-Jordá
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Plane Waves ◽  
Variational Solution ◽  
Radial Case

scholarly journals Exact eigenstates of a nanometric paraboloidal emitter and field emission quantities

2018 ◽  
Vol 474 (2214) ◽  
pp. 20170692 ◽  
Author(s):  
A. Chatziafratis ◽  
G. Fikioris ◽  
J. P. Xanthakis
Keyword(s):  
Schrödinger Equation ◽  
Field Emission ◽  
Schrodinger Equation ◽  
Near Field ◽  
Plane Waves ◽  
Spot Size ◽  
Emission Angle ◽  
Electron Velocity ◽  
Cartesian Geometry ◽  
Tip Radius

The progress in field emission theory from its initial Fowler–Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1–5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system. Significant differences from the usual Cartesian geometry are obtained. (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) The parallel electron velocity vanishes at the apex which may explain the recent spot-size measurements in near-field scanning electron microscopy. (5) Competing effects are found as the tip radius decreases to 1 nm: The electric field increases but the total supply function decreases so that possibly an optimum radius exists.

scholarly journals Solution of the Schrödinger torsion equation in the basis set of Mathieu functions: verification by numerical experiment

2021 ◽  
Vol 2052 (1) ◽  
pp. 012004
Author(s):  
A N Belov ◽  
V V Turovtsev ◽  
Yu A Fedina ◽  
Yu D Orlov
Keyword(s):  
Schrödinger Equation ◽  
Numerical Experiment ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Periodic Potential ◽  
Plane Waves ◽  
Basis Set ◽  
Basis Sets ◽  
Mathieu Functions ◽  
Computational Stability

Abstract The efficiency of the algorithm for the numerical solution of the Schrödinger torsion equation in the basis of Mathieu functions has been considered. The computational stability of the proposed algorithm is shown. The energies of torsion transitions determined in the basis sets of plane waves and Mathieu functions have been compared with the results of spectroscopy. A conclusion about the applicability of the algorithm using the basis set of Mathieu functions to the solution of the Schrödinger equation with a periodic potential has been derived.

scholarly journals Soliton and Breather Splitting on Star Graphs from Tricrystal Josephson Junctions

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 271
Author(s):  
Hadi Susanto ◽  
Natanael Karjanto ◽  
Zulkarnain ◽  
Toto Nusantara ◽  
Taufiq Widjanarko
Keyword(s):  
Schrödinger Equation ◽  
Josephson Junctions ◽  
Small Amplitude ◽  
Schrodinger Equation ◽  
Gordon Equation ◽  
Complex Dynamics ◽  
Transmission Rate ◽  
Plane Waves ◽  
Star Graph ◽  
Sine Gordon Equation

We consider the interactions of traveling localized wave solutions with a vertex in a star graph domain that describes multiple Josephson junctions with a common/branch point (i.e., tricrystal junctions). The system is modeled by the sine-Gordon equation. The vertex is represented by boundary conditions that are determined by the continuity of the magnetic field and vanishing total fluxes. When one considers small-amplitude breather solutions, the system can be reduced into the nonlinear Schrödinger equation posed on a star graph. Using the equation, we show that a high-velocity incoming soliton is split into a transmitted component and a reflected one. The transmission is shown to be in good agreement with the transmission rate of plane waves in the linear Schrödinger equation on the same graph (i.e., a quantum graph). In the context of the sine-Gordon equation, small-amplitude breathers show similar qualitative behaviors, while large-amplitude ones produce complex dynamics.

DISORDER IN THE DISCRETE NONLINEAR SCHRÖDINGER EQUATION

1995 ◽  
Vol 09 (16) ◽  
pp. 1899-1932 ◽  
Author(s):  
M.I. MOLINA ◽  
G.P. TSIRONIS
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Disordered Systems ◽  
Schrodinger Equation ◽  
Plane Waves ◽  
Stationary Problem ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinearity Parameter ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger

We study the interplay of disorder and nonlinearity in condensed matter systems modeled by the paradigmatic discrete nonlinear Schrödinger equation and focus on selftrapping and transport properties. We start by analyzing the two most simple nonlinear disordered systems, viz. a nondegenerate nonlinear dimer and a perturbed degenerate dimer. We then consider the case of few nonlinear impurities embedded in a linear host and treat the stationary problem analytically. We conclude by examining the case of a one-dimensional nonlinear random binary alloy where we find absence of quasiparticle localization, except for large nonlinearity parameter values. The transmission of plane’ waves across a disordered nonlinear segment of this type shows a power-law decay as a function of the segment size.

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