Non-linear dynamics of relativistic particles: How good is the classical phase space approach?
20
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
-
Upload
winchell-vance -
Category
Documents
-
view
30 -
download
1
description
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 - PowerPoint PPT Presentation
Transcript of 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
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
Is classical mechanics valid in systems which
are non-linear due to relativistic speeds
Solution strategyCompare classical relativistic Liouville density
with the Quantum Dirac probability
??
Theoretical Approaches
• RK-4 variable step size
it ic AV(r)Dirac
Liouville
Braun, Su, Grobe, PRA 59, 604 (1999)
Peverly, Wagner, Su, Grobe, Las Phys. 10, 303 (2000)
H c4 c2 (p A / c)2 V(r)
t
Pcl (x,p, t)H
x
p
Pcl H
p
x
Pcl
Construction of classical density distributions
|(r)|2
Large density
Quantumprobability
Classical particles
Classicaldensity
P(r)
choose wisely:
Pclr , t 1
N
1
22 3/2 exp r
r n(t) 2
22
n1
N
if too small: if too large:
Construction of a classical density
0
0.5
1
1.5
2
0.0001 0.001 0.01 0.1 1
Width of each mini-gaussian
% Error(constant width )
0
5.2
10.4
15.6
20.8
10 100 1000 104 105
Number of mini-gaussians N
% Error(constant N)
Accuracy optimization
Relativistic 1D harmonic oscillator
V(x) 1
20
2x2A cE0
Lsin Lt
H. Kim, M. Lee, J. Ji, J. Kim, PRA 53, 3767 (1996)
• simplest system to study relativity for classical and quantum theories
• dynamics can be chaotic
Wagner, Peverly, Su, Grobe, Phys. Rev. A 61, 35402 (2000)
Hmag
1
2p
1
c
A M(
r )
1
c
A L(t)
2
A M(
r )
c2
y
x
0
Hosc px
2
2
py2
2
1
2
2
2
x2 y2 xcos2
t
ysin
2
t
E(t)
osc exp iLzt / 2 exp ir A L(t) / c mag
See Robert Wagner’s talk (C6.10) at 15:48 today
Exploit Resonance
Wagner, Su, Grobe, Phys. Rev. Lett. 84, 3284 (2000)
0
0.2
0.4
0.6
0.8
1
6 8 10 12 14
100 %
80 %
60 %
40 %
20 %
Velocity/c
L
0
rel
Non-rel
0
1
2
3
-10 0 10 20
P(x
,t)
x [a.u.]
t = 0 c
t = 4 c
t = 8 c
(a)
Relativistic
1
2
3
-10 0 10 20
P(x
,t)
x [a.u.]
t = 0 c
t = 4 c
t = 8 c(b)
Non-Rel
Spatial probability density P(x,t)
Non-Relativistic
-20
-10
0
10
20
50 100 150 200
<x
> qm &
<x
>
cl
t [a.u.]
Relativistic
Position <x> qm and <x>cl
Liouville = Schrödinger
Liouville ≈ Dirac !
-20
-10
0
10
20
0 50 100 150 200
<x
> qm &
<x
>
cl
t [a.u.]
Spatial width <x>
0
4
8
12
0 20 40 60 80 100
< x
> [
a.u
.]
t [in laser periods]
(a)
2
4
6
8
10
43 44 45 46
< x
> [a.u
.]
t [in laser periods]
(b)
classicalclassical
quantumquantum
QuickTime™ and aGIF decompressor
are needed to see this picture.
QuickTime™ and aGIF decompressor
are needed to see this picture.
QuickTime™ and aGIF decompressor
are needed to see this picture.
0
0.2
0.4
0.6
-10 0 10 20
Pq
m(x
,t)
x [a.u.]
t = 300 c
New structures
0
0.2
0.4
0.6
-10 0 10 20
Pcl (x
,t)
x [a.u.]
classicalclassical
DiracDirac
10-30
10-25
10-20
10-15
10-10
10-5
100
-10 0 10 20
Pq
m(x,t
)
x [a.u.]
t = 300 c
Sharp localization
10-30
10-25
10-20
10-15
10-10
10-5
100
-10 0 10 20
Pcl (
x,t)
x [a.u.]
classicalclassical
DiracDirac
Summary- Phase space approach
valid in relativistic regime- Novel relativistic structures
localization- Implication: cycloatom
www.phy.ilstu.edu/ILP
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)
