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Non-linear dynamics of relativistic particles: How good is the classical phase space approach?

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. Acknowledgment. Undergrad researchers: R. Wagner, cycloatoms

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Non-linear dynamics of relativistic particles: How good is the classical phase space approach?

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  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 www.phy.ilstu.edu/ILP

  2. 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 oscillatorssame For non-linear forcesdifferent

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

  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 large: if s too small:

  8. Accuracy optimization 20.8 15.6 10.4 5.2 0 Number of mini-gaussians N 4 5 10 100 1000 10 10 2 1.5 1 0.5 0 0.0001 0.001 0.01 0.1 1 % Error (constant width s ) % Error (constant N) 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, 35402 (2000)

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

  11. Exploit Resonance Velocity/c 100 % Non-rel 80 % 60 % rel 40 % 20 % w 0 w L Wagner, Su, Grobe, Phys. Rev. Lett. 84, 3284 (2000)

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

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

  14. Spatial width <Dx> classical quantum

  15. New structures Dirac classical

  16. Sharp localization Dirac classical

  17. 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

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