10.4 Dynamics of a Driven Bounded Particle
353
and transform from parameters ( ˜
p, ˜
x, ˜
H , ˜
, ˜
ω, ˜
t) to the dimensionless parameters
(p, x, H, ,, ω 0 , t) by setting ˜
x = xa, ˜
p = p ¯
h
a , ˜
= ¯
h 2
2ma 3 , ˜
H = H ¯
h 2
2ma 2 , ˜
t = t
2ma 2
¯
h ,
and ˜
ω = ω 0
¯
h
2ma 2 . In terms of these dimensionless parameters, the Hamiltonian takes
the form
H = p
2
+ xcos(ω 0 t), for |x| < 1.
(10.31)
We can also write the Hamiltonian in terms of action-angle variables (J =
2|p|/π, θ = π(x + 1)/2) (see Appendix B), and obtain
H =
π 2 J 2
4
−
2
π 2
∞
ν=−∞
1
(2ν − 1) 2 cos((2ν − 1)θ − ω 0 t), for |x| < 1.
(10.32)
This system has an infinite number of primary nonlinear resonances. From the
resonance condition ˙
θ =
∂H
∂J = π 2 J /2, the resonances are located at J =
J ν ≡2ω 0 /((2ν − 1)π 2 ). Note that as ν→∞, J ν →0 and the resonances accumulate
at low energies.
The Schrödinger equation can be written in the position basis in terms of
dimensionless units if we note that the momentum operator is p = −i
∂
∂x . The
Schrödinger equation is
i
∂
∂t
x|ψ(t) =
−
∂ 2
∂x 2 + xcos(ω 0 t)
x|ψ(t),
(10.33)
where we require that x|ψ(t) = 0 for x = ±1 at all times t due to the hard walls
of the square-well system. In the limit →0, the energy is a good quantum number.
In that limit, the energy eigenvalues, E n , and orthonormal energy eigenfunctions,
x|φ n = φ n (x), of the square-well system, in dimensionless units, are given by
E n =
n 2 π 2
4
and φ n (x) = sin
nπ(x − 1)
2
,
(10.34)
respectively. Note that for integer values of J the classical and quantum energies
coincide.
In the presence of a time-periodic driving field, this system has an infinite number
of primary nonlinear resonances (see Eq. (10.32)), and for each primary resonance
only an odd number of islands can occur. From the resonance condition ˙
θ =
∂H
∂J =
π 2 J /2, the resonances are located at J = J ν ≡2ω 0 /((2ν − 1)π 2 ). As ν→∞, J ν →0
and the resonances accumulate at low energies. A strobe plot of the classical phase
space for increasing values of is given in Fig. 10.5. With increasing amplitude of
the driving field, increasing numbers of nonlinear resonances overlap and lead to
full chaos at low energies.
353
and transform from parameters ( ˜
p, ˜
x, ˜
H , ˜
, ˜
ω, ˜
t) to the dimensionless parameters
(p, x, H, ,, ω 0 , t) by setting ˜
x = xa, ˜
p = p ¯
h
a , ˜
= ¯
h 2
2ma 3 , ˜
H = H ¯
h 2
2ma 2 , ˜
t = t
2ma 2
¯
h ,
and ˜
ω = ω 0
¯
h
2ma 2 . In terms of these dimensionless parameters, the Hamiltonian takes
the form
H = p
2
+ xcos(ω 0 t), for |x| < 1.
(10.31)
We can also write the Hamiltonian in terms of action-angle variables (J =
2|p|/π, θ = π(x + 1)/2) (see Appendix B), and obtain
H =
π 2 J 2
4
−
2
π 2
∞
ν=−∞
1
(2ν − 1) 2 cos((2ν − 1)θ − ω 0 t), for |x| < 1.
(10.32)
This system has an infinite number of primary nonlinear resonances. From the
resonance condition ˙
θ =
∂H
∂J = π 2 J /2, the resonances are located at J =
J ν ≡2ω 0 /((2ν − 1)π 2 ). Note that as ν→∞, J ν →0 and the resonances accumulate
at low energies.
The Schrödinger equation can be written in the position basis in terms of
dimensionless units if we note that the momentum operator is p = −i
∂
∂x . The
Schrödinger equation is
i
∂
∂t
x|ψ(t) =
−
∂ 2
∂x 2 + xcos(ω 0 t)
x|ψ(t),
(10.33)
where we require that x|ψ(t) = 0 for x = ±1 at all times t due to the hard walls
of the square-well system. In the limit →0, the energy is a good quantum number.
In that limit, the energy eigenvalues, E n , and orthonormal energy eigenfunctions,
x|φ n = φ n (x), of the square-well system, in dimensionless units, are given by
E n =
n 2 π 2
4
and φ n (x) = sin
nπ(x − 1)
2
,
(10.34)
respectively. Note that for integer values of J the classical and quantum energies
coincide.
In the presence of a time-periodic driving field, this system has an infinite number
of primary nonlinear resonances (see Eq. (10.32)), and for each primary resonance
only an odd number of islands can occur. From the resonance condition ˙
θ =
∂H
∂J =
π 2 J /2, the resonances are located at J = J ν ≡2ω 0 /((2ν − 1)π 2 ). As ν→∞, J ν →0
and the resonances accumulate at low energies. A strobe plot of the classical phase
space for increasing values of is given in Fig. 10.5. With increasing amplitude of
the driving field, increasing numbers of nonlinear resonances overlap and lead to
full chaos at low energies.