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 www.phy.ilstu.edu/ILP

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

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

? 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 ?

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

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

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

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

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, 35402 (2000)

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

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)

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

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

Spatial width <Dx> classical quantum

New structures Dirac classical

Sharp localization Dirac classical

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) www.phy.ilstu.edu/ILP