198
7 Bounded Quantum Systems
In Sect. 7.9 we show that the signatures of chaos can also be found in two
dimensional classical and quantum periodic lattices. We study the dynamical
behavior of a “soft-Sinai” square lattice, which is a square lattice with Gaussian
potential peaks at the center of each unit cell of the lattice (Porter et al. 2017). We
use a lattice surface of section (LSOS) to show that, near the top of the potential
energy peaks the classical dynamics is almost fully chaotic. A LSOS is a plot of the
Birkhoff coordinates of a particle each time it crosses a given line in any unit cell of
the lattice as it traverses the lattice. The band structure of lattice was also shown to
exhibit signatures of chaos using the Peres test and a significant increase in avoided
crossings in regions of classical chaos.
Finally, in Sect. 7.10 we make some concluding remarks.
7.2 Quantum Integrability
A classical conservative system with N degrees of freedom is integrable if there
exist N independent global functions whose mutual Poisson brackets vanish.
Integrability in quantum systems is defined in an analogous manner. A quantum
system with N degrees of freedom is integrable if there exist N independent globally
defined operators, ˆ
I m ( ˆ
p 1 , . . . , ˆ
p N ; ˆ
q 1 , . . . , ˆ
q N ), for m = 1, . . . ,N, whose mutual
commutators vanish,
[ ˆ
I m , ˆ
I n ] = 0,
(7.1)
for all m, n = 1, . . . N (Zaslavsky 1981; Hietarinta 1982, 1984, 1998; Eckhardt
1988). Equation (7.1) implies that common eigenstates of the operators ˆ
I m (m =
1, . . . N) can be found.
A question of considerable interest is whether systems that are integrable
classically are also integrable quantum mechanically and conversely. Hietarinta
(1982, 1984) has used the Moyal bracket to do a systematic search for quantum
integrals of motion.
The Moyal bracket provides an expression for the commutator of two operators
in terms of a function of differential operators acting on classical phase functions
(see Appendix I). Hietarinta considered quantum systems with two degrees of
freedom whose classical counterparts are known to be classically integrable and
have Hamiltonians of the type
ˆ
H =
1
2
ˆ
p
2
1 +
1
2
ˆ
p
2
2 + V ( ˆ
q 1 , ˆ
q 2 ),
(7.2)
where V ( ˆ
q 1 , ˆ
q 2 ) is the potential energy operator. If the system is integrable, then
there exists an operator, ˆ
I , that is independent of ˆ
H such that the phase functions,
H (p 1 , p 2 , q 1 , q 2 ) and I (p 1 , p 2 , q 1 , q 2 ), associated with the operators ˆ
H and ˆ
I
satisfy the Moyal bracket condition
7 Bounded Quantum Systems
In Sect. 7.9 we show that the signatures of chaos can also be found in two
dimensional classical and quantum periodic lattices. We study the dynamical
behavior of a “soft-Sinai” square lattice, which is a square lattice with Gaussian
potential peaks at the center of each unit cell of the lattice (Porter et al. 2017). We
use a lattice surface of section (LSOS) to show that, near the top of the potential
energy peaks the classical dynamics is almost fully chaotic. A LSOS is a plot of the
Birkhoff coordinates of a particle each time it crosses a given line in any unit cell of
the lattice as it traverses the lattice. The band structure of lattice was also shown to
exhibit signatures of chaos using the Peres test and a significant increase in avoided
crossings in regions of classical chaos.
Finally, in Sect. 7.10 we make some concluding remarks.
7.2 Quantum Integrability
A classical conservative system with N degrees of freedom is integrable if there
exist N independent global functions whose mutual Poisson brackets vanish.
Integrability in quantum systems is defined in an analogous manner. A quantum
system with N degrees of freedom is integrable if there exist N independent globally
defined operators, ˆ
I m ( ˆ
p 1 , . . . , ˆ
p N ; ˆ
q 1 , . . . , ˆ
q N ), for m = 1, . . . ,N, whose mutual
commutators vanish,
[ ˆ
I m , ˆ
I n ] = 0,
(7.1)
for all m, n = 1, . . . N (Zaslavsky 1981; Hietarinta 1982, 1984, 1998; Eckhardt
1988). Equation (7.1) implies that common eigenstates of the operators ˆ
I m (m =
1, . . . N) can be found.
A question of considerable interest is whether systems that are integrable
classically are also integrable quantum mechanically and conversely. Hietarinta
(1982, 1984) has used the Moyal bracket to do a systematic search for quantum
integrals of motion.
The Moyal bracket provides an expression for the commutator of two operators
in terms of a function of differential operators acting on classical phase functions
(see Appendix I). Hietarinta considered quantum systems with two degrees of
freedom whose classical counterparts are known to be classically integrable and
have Hamiltonians of the type
ˆ
H =
1
2
ˆ
p
2
1 +
1
2
ˆ
p
2
2 + V ( ˆ
q 1 , ˆ
q 2 ),
(7.2)
where V ( ˆ
q 1 , ˆ
q 2 ) is the potential energy operator. If the system is integrable, then
there exists an operator, ˆ
I , that is independent of ˆ
H such that the phase functions,
H (p 1 , p 2 , q 1 , q 2 ) and I (p 1 , p 2 , q 1 , q 2 ), associated with the operators ˆ
H and ˆ
I
satisfy the Moyal bracket condition