This document studies how Coulomb repulsion and exchange interaction affect the time evolution and pulse duration of two-electron systems. The researchers construct an initial wavefunction for the two-electron system and time-evolve it using the Hamiltonian, which includes terms for both Coulomb repulsion and exchange interaction. Their results show that both effects contribute to increasing the separation between electrons over time, but are within an order of magnitude of each other, suggesting both play a role in limiting ultrafast electron pulse durations.
Traveling EM waves represent freely propagating energy. Standing waves represent bottled-up energy. Light is a traveling wave disturbance in a polarizable vacuum. Matter consists of standing wave resonances.
Matter in motion with respect to an inertial frame generates de Broglie matter waves (contracted moving standing waves). Rest mass and inertia result from confinement of electromagnetic radiation.
Traveling EM waves represent freely propagating energy. Standing waves represent bottled-up energy. Light is a traveling wave disturbance in a polarizable vacuum. Matter consists of standing wave resonances.
Matter in motion with respect to an inertial frame generates de Broglie matter waves (contracted moving standing waves). Rest mass and inertia result from confinement of electromagnetic radiation.
Electron Diffusion and Phonon Drag Thermopower in Silicon NanowiresAI Publications
The field of thermoelectric research has undergone a renaissance and boom in the fast two decades, largely fueled by the prospect of engineering electronic and phononic properties in nanostructures, among which semiconductor nanowires (NWs) have served both as an important platform to investigate fundamental thermoelectric transport phenomena and as a promising route for high thermoelectric performance for device applications. In this report we theoretical studied the carrier diffusion and phonon-drag contribution to thermoelectric performance of silicon nanowires and compared with the existing experimental data. We observed a good agreement between theoretical data and experimental observations in the overall temperature range from 50 – 350 K. Electron diffusion thermopower is found to be dominant mechanism in the low temperature range and shows linear dependence with temperature.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
Electron Diffusion and Phonon Drag Thermopower in Silicon NanowiresAI Publications
The field of thermoelectric research has undergone a renaissance and boom in the fast two decades, largely fueled by the prospect of engineering electronic and phononic properties in nanostructures, among which semiconductor nanowires (NWs) have served both as an important platform to investigate fundamental thermoelectric transport phenomena and as a promising route for high thermoelectric performance for device applications. In this report we theoretical studied the carrier diffusion and phonon-drag contribution to thermoelectric performance of silicon nanowires and compared with the existing experimental data. We observed a good agreement between theoretical data and experimental observations in the overall temperature range from 50 – 350 K. Electron diffusion thermopower is found to be dominant mechanism in the low temperature range and shows linear dependence with temperature.
In computational physics and Quantum chemistry, the Hartree–Fock (HF) method also known as self consistent method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system or many electron system in a stationary state
STRUCTURE OF ATOM
Sub atomic Particles
Atomic Models
Atomic spectrum of hydrogen atom:
Photoelectric effect
Planck’s quantum theory
Heisenberg’s uncertainty principle
Quantum Numbers
Rules for filling of electrons in various orbitals
1. Coulomb repulsion and exchange interaction
in dynamical two-electron systems
M. Gregoire, P. Lougovski, H. Batelaan – University of Nebraska-Lincoln
|cψ
(E)|^2 for l=1, m=-1,0,1is
Funding: NSF 2505210148001
Motivation 2:
What contributes most to the pulse duration limits of ultrafast
electron pulses, exchange interaction or Coulomb repulsion?
Motivation 1:
Is the free electron anti-correlation observed by Hasselbach
due to exchange interaction or Coulomb repulsion?
Tungsten tip
coherent illumination
Anode
Magnifying
quadrupole
doublet
Collector
anodes
MCP
- - - incoherent emission
coherent emission
Experiment: Results:
Hasselbach, Nature, 2002
Electron source:
Hommelhoff, Nature, 2011
Previous work on this subject:
Kasevich, PRL, 2006
Electron pulse compression:
Zewail, Batelaan, PNAS, 2009
Construct initial wavefunction
Use the single-particle state
to construct spatially anti-symmetric state:
Then use the coordinate substitution
to get initial wavefunction as a product of a center-of-
mass-coordinates part and a relative-coordinates part:
1
2
CoM
x,y, or z
|Χ(R)|^2
center-of-mass part:
3D Gaussian peak
Cylindrical plot of r and θ
for φ=0:
0<r<rmax
, 0<θ<π
Cylindrical plot of r and φ
for θ=0:
0<r<rmax
, 0<φ<2π
|ψ(R)|^2 |ψ(R)|^2
relative-coords part:
2 3D Gaussian peaks
Time-evolve initial wavefunction
The Hamiltonian of the system is:
Expand in E basis by finding expansion coefficients:
where is the Coulomb Wavefunction.
For :
Reconstruct wavefunction for arbitrary time t:
Results
Time evolution:
For initial conditions of two electrons emitted from a field
emission tip, we plot the expected value of r over time:
We extrapolate <r> to t=5 ns to estimate the differences
in arrival times between electrons for each case:
The effects of Coulomb repulsion and exchange
interaction are within one order of magnitude.
<r> at t=5 ns arrival time difference
with only exchange interaction 2.1∙10-5
1.8∙10-12
with exchange interaction and
Coulomb repulsion
3.0∙10-5
2.6∙10-12
|Χ(R)|^2
x
|Χ(R)|^2
y or z
|ψ(R)|^2 |ψ(R)|^2
|ψ(R)|^2
|ψ(R)|^2
Cylindrical
plots of r
and θ for
φ=0:
0<r<rmax
,
0<θ<π
Cylindrical
plots of r
and φ for
θ=0:
0<r<rmax
,
0<φ<2π
