Quantized Rotational Motion in a Plane

2020 ◽  
pp. 306-312
Author(s):  
Jochen Autschbach
Keyword(s):  
Angular Momentum ◽  
Boundary Conditions ◽  
Harmonic Oscillator ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Hindered Rotation ◽  
Methyl Group ◽  
Low Energies ◽  
Fixed Radius ◽  
The One

The angular momentum for the simplified case of a particle rotating in a fixed plane is treated. The ‘perimeter model’ is the analogue of the one-dimensional particle in a box (PiaB), with the particle moving on a circle with fixed radius. This requires cyclic – or periodic – boundary conditions. It is shown that the quantum perimeter model results can be obtained by re-interpreting the coordinate of the linear PiaB and by considering the periodic boundary conditions. The eigenvalue pattern leads to a 4n+2 Huckel rule. Next, the chapter discusses hindered rotations, such as the rotation of a methyl group around a C-C bond. The solutions to the hindered rotation problem combine features of the harmonic oscillator at low energies, with features of the perimeter model at high energies.

scholarly journals Must be the wave function single-valued? Ring vs. periodic boundary conditions, spinors and double rotations

2022 ◽  
Author(s):  
Josep Planelles
Keyword(s):  
Wave Function ◽  
Angular Momentum ◽  
Boundary Conditions ◽  
Undergraduate Students ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Unitary Groups ◽  
Lecture Notes

This is a lecture notes for undergraduate students. We try to tackle the single valuedness of spatial and double valuedness of spin functions. Also, we adress the need of spinors to accommodate spin functions with some parallelism to the need of axial vectors (or antisymmetric traceless tensors) to accommodate angular momentum. Finally, we revisit the Dirac and Weyl tricks on the non-equivalence of a 2 pi and a 4 pi rotation related the topology of rotation and unitary groups.

Inverse cascades of angular momentum

1996 ◽  
Vol 56 (3) ◽  
pp. 615-639 ◽  
Author(s):  
Shuojun Li ◽  
David Montgomery ◽  
Wesley B. Jones
Keyword(s):  
Angular Momentum ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Galerkin Approximation ◽  
Periodic Boundary Conditions ◽  
Navier Stokes ◽  
Box Dimension ◽  
Physical Situation ◽  
Two Dimensional ◽  
Decaying Turbulence

Most theoretical and computational studies of turbulence in Navier—Stokes fluids and/or guiding-centre plasmas have been carried out in the presence of spatially periodic boundary conditions. In view of the frequently reproduced result that two-dimensional and/or MHD decaying turbulence leads to structures comparable in length scale to a box dimension, it is natural to ask if periodic boundary conditions are an adequate representation of any physical situation. Here, we study, computationally, the decay of two-dimensional turbulence in a Navier—Stokes fluid or guiding-centre plasma in the presence of circular no-slip rigid walls. The method is wholly spectral, and relies on a Galerkin approximation by a set of functions that obey two boundary conditions at the wall radius (analogues of the Chandrasekhar— Reid functions). It is possible to explore Reynolds numbers up to the order of 1250, based on an RMS velocity and a box radius. It is found that decaying turbulence is altered significantly by the no-slip boundaries. First, strong boundary layers serve as sources of vorticity and enstrophy and enhance the early-time energy decay rate, for a given Reynolds number, well above the periodic boundary condition values. More importantly, angular momentum turns out to be an even more slowly decaying ideal invariant than energy, and to a considerable extent governs the dynamics of the decay. Angular momentum must be taken into account, for example, in order to achieve quantitative agreeement with the predicition of maximum entropy, or ‘most probable’, states. These are predicitions of conditions that are established after several eddy turnover times but before the energy has decayed away. Angular momentum will cascade to lower azimuthal mode numbers, even if absent there initially, and the angular momentum modal spectrum is eventually dominated by the lowest mode available. When no initial angular momentum is present, no behaviour that suggests the likelihood of inverse cascades is observed.

scholarly journals PERIODIC SOLUTIONS OF THE VLASOV–POISSON SYSTEM WITH BOUNDARY CONDITIONS

2000 ◽  
Vol 10 (05) ◽  
pp. 651-672 ◽  
Author(s):  
M. BOSTAN ◽  
F. POUPAUD
Keyword(s):  
Boundary Conditions ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Space Dimension ◽  
Periodic Boundary Conditions ◽  
Small Data ◽  
Poisson System ◽  
One Dimensional ◽  
The One ◽  
Time Periodic

We study the Vlasov–Poisson system with time periodic boundary conditions. For small data we prove existence of weak periodic solutions in any space dimension. In the one-dimensional case the result is stronger: we obtain existence of mild solution and uniqueness of this solution when the data are smooth. It is necessary to impose a nonvanishing condition for the incoming velocities in order to control the lifetime of particles in the domain.

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

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