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7 Bounded Quantum Systems
Fig. 7.4 (a) The periodic orbit composed of repetitions of the sequence (0,0,1). (b) and (c) Show
contour plots of the Husimi distribution of probability for two of the eigenstates of the quantized
baker’s map, ¯
B S , that show scarring from the periodic orbit (0,0,1). The black regions show the
distribution at 70% of maximum, and contour lines at 50 and 30% probability have been drawn
(Saraceno 1990)
Fig. 7.5 (a) The periodic orbit composed of repetitions of the sequence (0,0,0,1,1,1). (b) and
(c) show contour plots of the Husimi distribution of probability for two of the eigenstates of the
quantized baker’s map, ¯
B S , that show scarring from the periodic orbit (0,0,0,1,1,1). The black
regions show the distribution at 70% of maximum, and contour lines at 50 and 30% probability
have been drawn (Saraceno 1990)
ˆ
q|n = n
2π ¯
h
L p
|n and ˆ
p|k = k
2π ¯
h
L q
|k
(7.10)
where n and k are integers, n = 0, 1, . . . , N − 1 and k = 0, 1, . . . , N − 1. The
quantity is
=
1
√
N
exp
i
2πnk
N
.
(7.11)
An arbitrary state of the system, | can be written in the position representation as
= =n| L R , where the states | L and | R are defined as L =
0 for n ≥ N and R = 0 for n ≤ N − 1. Similarly, we can write =