1. Non-linear dynamics of relativistic particles: How good is the classical phase space approach?
Peter J. Peverly
Sophomore
Intense Laser Physics Theory Unit
Illinois State University

2. National Science Foundation
Acknowledgment
Undergrad researchers:
R. Wagner, cycloatoms
J. Braun, quantum simulations
A. Bergquist, graphics
T. Shepherd, animations
Advisors:
Profs. Q. Su, R. Grobe
Support:
National Science Foundation
Research Corporation
ISU Honor’s Program

3. Quantum probabilities vs
classical distributions
For harmonic oscillators same
For non-linear forces different

4. ? Motivation Solution strategy
Is classical mechanics valid in systems which
are non-linear due to relativistic speeds
Solution strategy
Compare classical relativistic Liouville density
with the Quantum Dirac probability ?

5. Theoretical Approaches
Dirac
Braun, Su, Grobe, PRA 59, 604 (1999)
Liouville
Peverly, Wagner, Su, Grobe, Las Phys. 10, 303 (2000)
RK-4 variable step size

6. Construction of classical density distributions
Quantum
probability
|Y(r)|2
Classical
particles
Large density
P(r)
Classical
density

7. Construction of a classical density
choose s wisely:
if s too small:
if s too large:

8. Accuracy optimization
5.2
10.4
15.6
20.8 10
100
1000 4 5
Number of mini-gaussians N
% Error
(constant width s )
0.5 1
1.5 2
% Error
(constant N)
0.0001
0.001
0.01
0.1 1
Width of each mini-gaussian s

9. Relativistic 1D harmonic oscillator
simplest system to study relativity for classical and quantum theories
dynamics can be chaotic
H. Kim, M. Lee, J. Ji, J. Kim, PRA 53, 3767 (1996)
Wagner, Peverly, Su, Grobe, Phys. Rev. A 61, (2000)

10. Nonrel electron in B-field = rotating 2D oscillator
See Robert Wagner’s talk (C6.10) at 15:48 today

11. Exploit Resonance Non-rel rel w w Velocity/c 100 % 80 % 60 % 40 % 20 %w L
Wagner, Su, Grobe, Phys. Rev. Lett. 84, 3284 (2000)

12. Spatial probability density P(x,t)
Non-Rel
Relativistic

13. Position <x> qm and <x>cl
Non-Relativistic
Liouville = Schrödinger
Relativistic
Liouville ≈ Dirac !

14. Spatial width <Dx>
classical
quantum

18. New structures
Dirac
classical

19. Sharp localization
Dirac
classical

20. Summary - Phase space approach valid in relativistic regime
- Novel relativistic structures
localization
- Implication: cycloatom
Peverly, Wagner, Su, Grobe, Las. Phys. 10, 303 (2000)
Wagner, Peverly, Su, Grobe, Phys. Rev. A 61, 3502 (2000)
Su, Wagner, Peverly, Grobe, Front. Las. Phys. 117 (Springer, 2000)