Chapter 7
Bounded Quantum Systems
Abstract Classical chaos manifests itself in the quantum dynamics of bounded
systems (discrete energy spectrum) through the statistical properties of the energy
spectrum and the eigenvector spatial distributions. Two-dimensional quantum billiards (circle billiard, stadium billiard, Sinai billiard, and ripple billiard) have
been especially important for studies of the manifestation of chaos in quantum
dynamics. These systems show Wigner distributions of energy level spacing
and eigenstate scarring in numerical experiments and in laboratory experiments
involving microwave cavities.
Anharmonic oscillators are bounded quantum systems that can exhibit a transition to chaos. A one-dimensional triatomic molecule with anharmonic interatomic
coupling shows classical and quantum chaos. The signatures of chaos can also be
found in two dimensional classical and quantum periodic lattices. A example is the
“soft-Sinai” square lattice, which is a square lattice with Gaussian potential peaks
at the center of each unit cell of the lattice. A generalization of Poincare surface
of section to periodic lattices (the lattice surface of section) shows that, near the
top of the potential energy peaks the classical dynamics is almost fully chaotic. The
band structure of the lattice shows signatures of chaos and a significant increase in
avoided crossings in regions of classical chaos.
Keywords Bounded quantum systems · Random matrix theory · Wigner
distribution · Integrable quantum system · Moyal bracket · Diabolical points ·
Quantum baker’s map · Sinai billiard · Stadium billiard · Ripple billiard · Peres
test for chaos · Anharmonic oscillators · Sinai lattice · Lattice surface of section
7.1 Introduction
In this chapter, we study the effects of classical chaos on the quantum dynamics
of conservative bounded systems (which have a discrete energy spectrum). In
subsequent chapters, we will consider open quantum systems (systems that have
a continuous energy spectrum) in which energy is conserved, and we will consider
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_7
195
Bounded Quantum Systems
Abstract Classical chaos manifests itself in the quantum dynamics of bounded
systems (discrete energy spectrum) through the statistical properties of the energy
spectrum and the eigenvector spatial distributions. Two-dimensional quantum billiards (circle billiard, stadium billiard, Sinai billiard, and ripple billiard) have
been especially important for studies of the manifestation of chaos in quantum
dynamics. These systems show Wigner distributions of energy level spacing
and eigenstate scarring in numerical experiments and in laboratory experiments
involving microwave cavities.
Anharmonic oscillators are bounded quantum systems that can exhibit a transition to chaos. A one-dimensional triatomic molecule with anharmonic interatomic
coupling shows classical and quantum chaos. The signatures of chaos can also be
found in two dimensional classical and quantum periodic lattices. A example is the
“soft-Sinai” square lattice, which is a square lattice with Gaussian potential peaks
at the center of each unit cell of the lattice. A generalization of Poincare surface
of section to periodic lattices (the lattice surface of section) shows that, near the
top of the potential energy peaks the classical dynamics is almost fully chaotic. The
band structure of the lattice shows signatures of chaos and a significant increase in
avoided crossings in regions of classical chaos.
Keywords Bounded quantum systems · Random matrix theory · Wigner
distribution · Integrable quantum system · Moyal bracket · Diabolical points ·
Quantum baker’s map · Sinai billiard · Stadium billiard · Ripple billiard · Peres
test for chaos · Anharmonic oscillators · Sinai lattice · Lattice surface of section
7.1 Introduction
In this chapter, we study the effects of classical chaos on the quantum dynamics
of conservative bounded systems (which have a discrete energy spectrum). In
subsequent chapters, we will consider open quantum systems (systems that have
a continuous energy spectrum) in which energy is conserved, and we will consider
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3_7
195