208
7 Bounded Quantum Systems
Some of Saraceno’s results are shown in Figs. 7.4, 7.5 and 7.6. In each of Figs. 7.4
and 7.5, we show two eigenstates of ¯
B S that show scars of the classical orbit in that
figure. In Fig. 7.6b, we show an eigenstate of ¯
B S that shows scars of the heteroclinic
orbit in Fig. 7.6a.
The quantum baker’s map has been used by a variety of authors to study issues
related to the semiclassical limit. Kaplan and Heller have used it to study the effect
of homoclinic recurrence on scar strength (Kaplan and Heller 1998). Soklakov and
Schack showed that the quantum baker’s map approaches a classical Bernoulli shift
in the limit of small Planck’s constant (Soklakov and Schack 2000). Several authors
have worked on the development of a semiclassical path integral for the quantum
baker’s map (O’Conner et al. 1992; Dittes et al. 1994; Eckhardt and Haake 1994;
Saraceno and Voros 1994; Luz and Ozorio de Almeida 1995; Kaplan and Heller
1996; Toscano et al. 1997; Tanner 1999).
There have also been some studies that discuss how the quantum baker’s map
could be realized by a quantum computer (Schack 1998; Brun and Schack 1999).
7.5 Quantum Billiards
Quantum billiards have played an important role in our understanding of the
quantum-classical correspondence for systems that can undergo a transition to
chaos, because their shapes are easily changed and the nature of the dynamics
can be controlled. Also, they can be totally closed to form bounded systems, or
they can form cavities in waveguides and, thereby, affect the dynamics of open
systems. Below, we describe the spectral statistics of billiards with several different
shapes, including the chaotic stadium billiard and Sinai billiard, and finally the
ripple billiard, which can have a chaotic or a mixed phase space.
7.5.1 The Stadium Billiard
The first convincing evidence of a strong connection between the spectral statistics
of a quantum system and the dynamics of its classical counterpart is due to
McDonald and Kaufman (1979), who studied the spectral statistics of a quantum
particle in the stadium billiard (see Fig. 2.17). Classically, the dynamics of a particle
in a circular stadium (a = 0 in Fig. 2.17) is integrable, while the dynamics of a
particle in a noncircular stadium (a =0) is chaotic.
McDonald and Kaufman computed histograms of energy eigenvalue nearest
neighbor spacings for both the classically integrable (a = 0) and classically chaotic
(a =0) cases. This was done by solving the Schrodinger equation for a free particle
with mass m and energy E,
7 Bounded Quantum Systems
Some of Saraceno’s results are shown in Figs. 7.4, 7.5 and 7.6. In each of Figs. 7.4
and 7.5, we show two eigenstates of ¯
B S that show scars of the classical orbit in that
figure. In Fig. 7.6b, we show an eigenstate of ¯
B S that shows scars of the heteroclinic
orbit in Fig. 7.6a.
The quantum baker’s map has been used by a variety of authors to study issues
related to the semiclassical limit. Kaplan and Heller have used it to study the effect
of homoclinic recurrence on scar strength (Kaplan and Heller 1998). Soklakov and
Schack showed that the quantum baker’s map approaches a classical Bernoulli shift
in the limit of small Planck’s constant (Soklakov and Schack 2000). Several authors
have worked on the development of a semiclassical path integral for the quantum
baker’s map (O’Conner et al. 1992; Dittes et al. 1994; Eckhardt and Haake 1994;
Saraceno and Voros 1994; Luz and Ozorio de Almeida 1995; Kaplan and Heller
1996; Toscano et al. 1997; Tanner 1999).
There have also been some studies that discuss how the quantum baker’s map
could be realized by a quantum computer (Schack 1998; Brun and Schack 1999).
7.5 Quantum Billiards
Quantum billiards have played an important role in our understanding of the
quantum-classical correspondence for systems that can undergo a transition to
chaos, because their shapes are easily changed and the nature of the dynamics
can be controlled. Also, they can be totally closed to form bounded systems, or
they can form cavities in waveguides and, thereby, affect the dynamics of open
systems. Below, we describe the spectral statistics of billiards with several different
shapes, including the chaotic stadium billiard and Sinai billiard, and finally the
ripple billiard, which can have a chaotic or a mixed phase space.
7.5.1 The Stadium Billiard
The first convincing evidence of a strong connection between the spectral statistics
of a quantum system and the dynamics of its classical counterpart is due to
McDonald and Kaufman (1979), who studied the spectral statistics of a quantum
particle in the stadium billiard (see Fig. 2.17). Classically, the dynamics of a particle
in a circular stadium (a = 0 in Fig. 2.17) is integrable, while the dynamics of a
particle in a noncircular stadium (a =0) is chaotic.
McDonald and Kaufman computed histograms of energy eigenvalue nearest
neighbor spacings for both the classically integrable (a = 0) and classically chaotic
(a =0) cases. This was done by solving the Schrodinger equation for a free particle
with mass m and energy E,