7.4 The Quantized Baker’s Map
203
and 7 at Y = 3.97 and satisfies the condition that the wave function changes sign
during a rotation of 360 ◦ in parameter space.
It is of interest to note that there has been a significant amount of work, inspired
by the work of Pechukas (1983), that studies avoided crossings by constructing
equations of motion for the eigenvalues as a function of some parameter such
as Planck’s constant or a coupling constant. Further discussion of this topic can
be found in Gibbons and Hermsen (1984), Gaspard et al. (1989), Nakamura
and Lakshmann (1987), Nakamura and Mikeska (1987), Stöckmann (1999), and
Yukawa (1985).
7.4 The Quantized Baker’s Map
The classical baker’s map is one of the simplest dynamical systems that is a Kflow. As discussed in Chap. 2, its phase space consists of the unit square, and the
points in its phase space are in one-to-one correspondence with all possible infinite
sequences that can be constructed from a two-letter alphabet consisting of 0 and 1.
The dynamical evolution occurs in discrete steps, and the orbit of any initial point is
obtained by shifting all entries in its sequence to the right by one place at each step.
The baker’s map contains many of the important features of more complicated
dynamical systems. It has an infinite number of periodic orbits. These correspond
to sequences that consist of an infinite repetition of a finite length sequence. Of
these periodic orbits, only two are period 1 fixed points. These occur at (p =
0, q = 0) and (p = 1, q = 1) and correspond to sequences with all 0’s or all
1’s, respectively. It has one period 2 orbit consisting of an infinite repetition of
the finite sequence (0,1). That is, (. . . ,0,1,0,1,0,1,. . . ). A period 3 and a period
6 orbit are shown in Figs. 7.4a and 7.5a, respectively. The baker’s map also has
homoclinic and heteroclinic orbits. These correspond to sequences that at one
end are periodic, consisting of a repetition of a finite sequence, γ (for example,
γ = (0, 1)), and then an excursion through some other finite sequence and a
return at the other end to an infinite repetition of γ for a homoclinic orbit or γ
for a heteroclinic orbit. For example, a homoclinic orbit might have a sequence
(. . . , γ, γ, 1, 1, 1, 1, 1, γ, γ, . . .). The heteroclinic orbit (. . . 0,0,0,1,0,1,1,1,. . . ) is
shown in Fig. 7.6a.
The baker’s map was first quantized by Balazs and Voros (1989), and we follow
their discussion here. Consider the baker’s map on a phase space square with area
L q L p (see Fig. 7.7). The number of quantum states that fit onto the square is N =
L q L p /2π ¯
h. We assume that N is even so that N = 2N , where N is an integer.
The width, q, of a single quantum state in the q-direction is q = 2π ¯
h/L p , and
the width, p, of a single quantum state in the p-direction is p = 2π ¯
h/L q . We
will denote an eigenstate of the position operator, ˆ
q, by |n and an eigenstate of the
momentum operator, ˆ
p, by |k, so that
203
and 7 at Y = 3.97 and satisfies the condition that the wave function changes sign
during a rotation of 360 ◦ in parameter space.
It is of interest to note that there has been a significant amount of work, inspired
by the work of Pechukas (1983), that studies avoided crossings by constructing
equations of motion for the eigenvalues as a function of some parameter such
as Planck’s constant or a coupling constant. Further discussion of this topic can
be found in Gibbons and Hermsen (1984), Gaspard et al. (1989), Nakamura
and Lakshmann (1987), Nakamura and Mikeska (1987), Stöckmann (1999), and
Yukawa (1985).
7.4 The Quantized Baker’s Map
The classical baker’s map is one of the simplest dynamical systems that is a Kflow. As discussed in Chap. 2, its phase space consists of the unit square, and the
points in its phase space are in one-to-one correspondence with all possible infinite
sequences that can be constructed from a two-letter alphabet consisting of 0 and 1.
The dynamical evolution occurs in discrete steps, and the orbit of any initial point is
obtained by shifting all entries in its sequence to the right by one place at each step.
The baker’s map contains many of the important features of more complicated
dynamical systems. It has an infinite number of periodic orbits. These correspond
to sequences that consist of an infinite repetition of a finite length sequence. Of
these periodic orbits, only two are period 1 fixed points. These occur at (p =
0, q = 0) and (p = 1, q = 1) and correspond to sequences with all 0’s or all
1’s, respectively. It has one period 2 orbit consisting of an infinite repetition of
the finite sequence (0,1). That is, (. . . ,0,1,0,1,0,1,. . . ). A period 3 and a period
6 orbit are shown in Figs. 7.4a and 7.5a, respectively. The baker’s map also has
homoclinic and heteroclinic orbits. These correspond to sequences that at one
end are periodic, consisting of a repetition of a finite sequence, γ (for example,
γ = (0, 1)), and then an excursion through some other finite sequence and a
return at the other end to an infinite repetition of γ for a homoclinic orbit or γ
for a heteroclinic orbit. For example, a homoclinic orbit might have a sequence
(. . . , γ, γ, 1, 1, 1, 1, 1, γ, γ, . . .). The heteroclinic orbit (. . . 0,0,0,1,0,1,1,1,. . . ) is
shown in Fig. 7.6a.
The baker’s map was first quantized by Balazs and Voros (1989), and we follow
their discussion here. Consider the baker’s map on a phase space square with area
L q L p (see Fig. 7.7). The number of quantum states that fit onto the square is N =
L q L p /2π ¯
h. We assume that N is even so that N = 2N , where N is an integer.
The width, q, of a single quantum state in the q-direction is q = 2π ¯
h/L p , and
the width, p, of a single quantum state in the p-direction is p = 2π ¯
h/L q . We
will denote an eigenstate of the position operator, ˆ
q, by |n and an eigenstate of the
momentum operator, ˆ
p, by |k, so that