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.

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