3.9 Conclusions
95
just before the nth collision with the wall and the phase of the wall, ψ n , at the nth
collision. A simplified version of the Fermi map (Lieberman and Lichtenberg 1972)
is given by
u n+1 =
u n + ψ n −
1
2
,
(3.103)
ψ n+1 = ψ n +
C
u n+1
(mod1),
(3.104)
where C is a constant that depends on the amplitude of the wall oscillations and
the average distance between the walls. This system can go from quasiperiodic to
chaotic behavior as the constant C is varied. However, the chaotic region is restricted
to low energy so the particle cannot attain infinitely high velocities.
There are several maps associated with perturbed Kepler systems. Because of the
long range of the Coulomb potential, the whisker map does not adequately describe
the stochastic layer of hydrogen. A generalized whisker map of the stochastic layer
(including the effect of a constant field of arbitrary size) has been derived by Cocke
and Reichl (1990).
Casati et al. (1988) have used the fact that the singularity in the Coulomb
potential at x = 0 has a kick-like effect on the motion to derive a map, the
Kepler map, for driven one-dimensional hydrogen that is generally valid when no
constant field (Stark field) is present and can be generalized to include a small
constant field. For the case of a classical one-dimensional hydrogen atom driven
by a monochromatic time-periodic external field with frequency ω 0 , the Kepler map
is a map of the number of photons absorbed, N j , and the change in phase, φ j , of the
external field for each orbit of the electron. It is given by
N j +1 = N j + κ sin(φ j ),
(3.105)
φ j +1 = φ j + 2πω 0 (−2ω 0 N j +1 )
−3/2 .
(3.106)
In (Casati et al. 1988) it is shown that the Kepler map gives a good description of
one-dimensional microwave-driven hydrogen as long as the frequency ω 0 is not too
low.
Another map related to a perturbed Kepler system is the cometary map derived
by Petrosky (1986). This map describes the mechanism by which comets that orbit
the sun may be captured or lost due to the perturbing influence of Jupiter. Petrosky’s
cometary map may be written
P n+1 = P n + K sin(g n ),
(3.107)
g n+1 = g n −
2π
(−P n+1 ) 3/2 ,
(3.108)
95
just before the nth collision with the wall and the phase of the wall, ψ n , at the nth
collision. A simplified version of the Fermi map (Lieberman and Lichtenberg 1972)
is given by
u n+1 =
u n + ψ n −
1
2
,
(3.103)
ψ n+1 = ψ n +
C
u n+1
(mod1),
(3.104)
where C is a constant that depends on the amplitude of the wall oscillations and
the average distance between the walls. This system can go from quasiperiodic to
chaotic behavior as the constant C is varied. However, the chaotic region is restricted
to low energy so the particle cannot attain infinitely high velocities.
There are several maps associated with perturbed Kepler systems. Because of the
long range of the Coulomb potential, the whisker map does not adequately describe
the stochastic layer of hydrogen. A generalized whisker map of the stochastic layer
(including the effect of a constant field of arbitrary size) has been derived by Cocke
and Reichl (1990).
Casati et al. (1988) have used the fact that the singularity in the Coulomb
potential at x = 0 has a kick-like effect on the motion to derive a map, the
Kepler map, for driven one-dimensional hydrogen that is generally valid when no
constant field (Stark field) is present and can be generalized to include a small
constant field. For the case of a classical one-dimensional hydrogen atom driven
by a monochromatic time-periodic external field with frequency ω 0 , the Kepler map
is a map of the number of photons absorbed, N j , and the change in phase, φ j , of the
external field for each orbit of the electron. It is given by
N j +1 = N j + κ sin(φ j ),
(3.105)
φ j +1 = φ j + 2πω 0 (−2ω 0 N j +1 )
−3/2 .
(3.106)
In (Casati et al. 1988) it is shown that the Kepler map gives a good description of
one-dimensional microwave-driven hydrogen as long as the frequency ω 0 is not too
low.
Another map related to a perturbed Kepler system is the cometary map derived
by Petrosky (1986). This map describes the mechanism by which comets that orbit
the sun may be captured or lost due to the perturbing influence of Jupiter. Petrosky’s
cometary map may be written
P n+1 = P n + K sin(g n ),
(3.107)
g n+1 = g n −
2π
(−P n+1 ) 3/2 ,
(3.108)