Energy and Momentum

2018 ◽  
Author(s):  
David M. Wittman
Keyword(s):  
Time Dilation ◽  
Speed Of Light ◽  
Massless Particles ◽  
The Everyday ◽  
Low Speeds ◽  
Energy And Momentum ◽  
Deep Relationship ◽  
Insight Into ◽  
Momentum Relation

Tis chapter explains the famous equation E = mc2 as part of a wider relationship between energy, mass, and momentum. We start by defning energy and momentum in the everyday sense. We then build on the stretching‐triangle picture of spacetime vectors developed in Chapter 11 to see how energy, mass, and momentum have a deep relationship that is not obvious at everyday low speeds. When momentum is zero (a mass is at rest) this energy‐momentum relation simplifes to E = mc2, which implies that mass at rest quietly stores tremendous amounts of energy. Te energymomentum relation also implies that traveling near the speed of light (e.g., to take advantage of time dilation for interstellar journeys) will require tremendous amounts of energy. Finally, we look at the simplifed form of the energy‐momentum relation when the mass is zero. Tis gives us insight into the behavior of massless particles such as the photon.

scholarly journals Scattering in Terms of Bohmian Conditional Wave Functions for Scenarios with Non-Commuting Energy and Momentum Operators

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 408
Author(s):  
Matteo Villani ◽  
Guillermo Albareda ◽  
Carlos Destefani ◽  
Xavier Cartoixà ◽  
Xavier Oriols
Keyword(s):  
Single Particle ◽  
Degrees Of Freedom ◽  
Single Photon ◽  
Open Quantum Systems ◽  
Wave Functions ◽  
Practical Application ◽  
Light Matter Interaction ◽  
Pure States ◽  
Energy And Momentum ◽  
Full Quantum

Without access to the full quantum state, modeling quantum transport in mesoscopic systems requires dealing with a limited number of degrees of freedom. In this work, we analyze the possibility of modeling the perturbation induced by non-simulated degrees of freedom on the simulated ones as a transition between single-particle pure states. First, we show that Bohmian conditional wave functions (BCWFs) allow for a rigorous discussion of the dynamics of electrons inside open quantum systems in terms of single-particle time-dependent pure states, either under Markovian or non-Markovian conditions. Second, we discuss the practical application of the method for modeling light–matter interaction phenomena in a resonant tunneling device, where a single photon interacts with a single electron. Third, we emphasize the importance of interpreting such a scattering mechanism as a transition between initial and final single-particle BCWF with well-defined central energies (rather than with well-defined central momenta).

scholarly journals Four-dimensional vibrational spectroscopy for nanoscale mapping of phonon dispersion in BN nanotubes

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Ruishi Qi ◽  
Ning Li ◽  
Jinlong Du ◽  
Ruochen Shi ◽  
Yang Huang ◽  
...  
Keyword(s):  
Boron Nitride ◽  
Energy Loss ◽  
Phonon Dispersion ◽  
Hexagonal Boron Nitride ◽  
Real Space ◽  
Boron Nitride Nanotubes ◽  
Detection Sensitivity ◽  
Optical Modes ◽  
Optical And Mechanical Properties ◽  
Energy And Momentum

AbstractDirectly mapping local phonon dispersion in individual nanostructures can advance our understanding of their thermal, optical, and mechanical properties. However, this requires high detection sensitivity and combined spatial, energy and momentum resolutions, thus has been elusive. Here, we demonstrate a four-dimensional electron energy loss spectroscopy technique, and present position-dependent phonon dispersion measurements in individual boron nitride nanotubes. By scanning the electron beam in real space while monitoring both the energy loss and the momentum transfer, we are able to reveal position- and momentum-dependent lattice vibrations at nanometer scale. Our measurements show that the phonon dispersion of multi-walled nanotubes is locally close to hexagonal-boron nitride crystals. Interestingly, acoustic phonons are sensitive to defect scattering, while optical modes are insensitive to small voids. This work not only provides insights into vibrational properties of boron nitride nanotubes, but also demonstrates potential of the developed technique in nanoscale phonon dispersion measurements.

ENERGY-LOSS EFFECT IN K-SHELL IONIZATION

1998 ◽  
Vol 08 (04) ◽  
pp. 225-233 ◽  
Author(s):  
TAKESHI MUKOYAMA
Keyword(s):  
Energy Loss ◽  
Cross Sections ◽  
Relativistic Correction ◽  
Correction Method ◽  
Exact Integration ◽  
Plane Wave Born Approximation ◽  
Ionization Cross Sections ◽  
Energy And Momentum ◽  
Shell Ionization ◽  
Loss Effect

The energy-loss effect of the projectile for direct inner-shell ionization cross sections by charged-particle impact has been examined. The relativistic and nonrelativistic calculations for K-shell ionization with and without the energy-loss effect are made in the plane-wave Born approximation and compared with the Brandt-Lapicki theory for the corrections of the relativistic and energy-loss effect. It is demonstrated that the Brandt-Lapicki method gives a good approximation to both relativistic and nonrelativistic cross sections, which implicitly take into account the energy-loss effect. However, the use of the Brandt-Lapicki relativistic correction method in the nonrelativistic theory with the exact integration limits for energy and momentum transfer overestimates the relativistic calculations for low-energy projectiles. This indicates that the Brandt-Lapicki method for correction of the electronic relativistic effect should be used only with their energy-loss correction method.

scholarly journals Linear Energy Density and the Flux of an Electric Field in Proca Tubes

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 640
Author(s):  
Vladimir Dzhunushaliev ◽  
Vladimir Folomeev ◽  
Abylaikhan Tlemisov
Keyword(s):  
Scalar Field ◽  
Electric Field ◽  
Higgs Field ◽  
Meissner Effect ◽  
Coupling Constant ◽  
Cylindrically Symmetric ◽  
Integral Characteristics ◽  
Energy And Momentum ◽  
Energy And Momentum Densities ◽  
Higgs Scalar

In this work, we study cylindrically symmetric solutions within SU(3) non-Abelian Proca theory coupled to a Higgs scalar field. The solutions describe tubes containing either the flux of a color electric field or the energy flux and momentum. It is shown that the existence of such tubes depends crucially on the presence of the Higgs field (there are no such solutions without this field). We examine the dependence of the integral characteristics (linear energy and momentum densities) on the values of the electromagnetic potentials at the center of the tube, as well as on the values of the coupling constant of the Higgs scalar field. The solutions obtained are topologically trivial and demonstrate the dual Meissner effect: the electric field is pushed out by the Higgs scalar field.

On Microfaceting Instability of Pt(110) Under Catalytic Oxidation of Adsorbed Co

1992 ◽  
Vol 263 ◽  
Author(s):  
M. Papoular
Keyword(s):  
Catalytic Oxidation ◽  
Oxidation Reaction ◽  
Nucleation And Growth ◽  
Point Of View ◽  
Momentum Balance ◽  
Layer By Layer ◽  
Adsorption Of Co ◽  
Specific Temperature ◽  
Energy And Momentum ◽  
Sawtooth Profile

ABSTRACTAs demonstrated by recent STM [1] and LEED [2] experiments the platinum (110) surface undergoes, at carbon monoxide submonolayer coverages, a phase transition from the 1 x 2 “missing-row” (reconstructed) state to the 1 x 1(bulk-like) state under specific temperature and partial-pressure conditions. The catalytic oxidation reaction CO + 1/2 → CO2 drives a microfaceting instability [3] [4] of the Pt(110) surface which ends up in a regular sawtooth profile with a period ≈ 200 Å, along the [110] direction.We introduce the idea that the rather extensive Pt mass transport, as involved in the process, could be energetically assisted by the reaction itself. Energy and momentum-balance considerations lead us to expect an energy ≲ 0.5 eV to be transferrable to thesubstrate. This should efficiently contribute to initiating the “scraping”process that leads to the microfaceted pattern.A simple model for nucleation and growth of facets is presented (see ref. 5), yielding characteristic times of order minutes (at T = 500 K), in fair agreement with experiment.Independently of the structural/catalytic problem, adsorption of CO at submonolayer coverages on, e.g., Pt(110) might be of interest from a surfactantphysics point of view (see ref. 6 for a very recent study on layer-by-layer homoepitaxial metal growth).

Energy and momentum relaxation of hot electrons in GaN/AlGaN

2002 ◽  
Vol 14 (13) ◽  
pp. 3457-3468 ◽  
Author(s):  
N Balkan ◽  
M C Arikan ◽  
S Gokden ◽  
V Tilak ◽  
B Schaff ◽  
...  
Keyword(s):  
Hot Electrons ◽  
Momentum Relaxation ◽  
Energy And Momentum

scholarly journals Energy‐ and momentum‐conserving model of splash entrainment in sand and snow saltation

2017 ◽  
Vol 44 (3) ◽  
pp. 1601-1609 ◽  
Author(s):  
Francesco Comola ◽  
Michael Lehning
Keyword(s):  
Energy And Momentum
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