202
7 Bounded Quantum Systems
γ = 10Y and β =
1 −
1
6
X
10Y + 10X,
where 0 ≤ X ≤ 9 and 0 ≤ Y ≤ 6. There are three cases for which this system
is known to be classically integrable, (α, β, γ ) = (60, 60, 60), (90, 45, 45), and
(90, 60, 30). For other cases, the system is classically chaotic. However, even when
the system is classically chaotic, there are special cases when spatial symmetries
exist. For example, when X = 9, the triangle is isosceles and has reflection
symmetry. In this case, the eigenstates will have either even or odd parity and the
spectrum is a mixed spectrum composed of two pure sequences, one due to even
parity states and the other due to odd parity states.
In Fig. 7.2b, we show the lowest 13 energy eigenstates for the case when X = 7
for the range 0.6 ≤ Y ≤ 6. For this range of angles, the system is classically
chaotic and no known symmetries exist except at Y = 6, where the system becomes
classically integrable. We see a number of avoided crossings in this figure but
no accidental degeneracies. At Y = 6, where the system is integrable, several
degeneracies occur. In Fig. 7.3a, we show the spectrum for the case X = 9. For this
value of X, the triangle is isosceles for all values of Y , and therefore, even though
the system is classically chaotic, a symmetry exists that causes the spectrum to form
a mixed sequence as discussed above. We see that a number of degeneracies occur
as we vary Y . These degeneracies are not accidental but due to the symmetry. Berry
and Wilkenson had to search hard for true accidental degeneracies. An example of
such a degeneracy is shown in Fig. 7.3b for X = 4.61. It occurs between levels 6
Fig. 7.3 Energy-level curves for the isosceles triangle. (a) X = 9: many level crossings occur
between states of different parity. The small boxes on the right indicate regions where more than
one degeneracy occurs. (b) X = 4.61: a true accidental degeneracy occurs between levels 6 and
7 at Y = 3.97. For this degeneracy, the wave function changes sign under a rotation in parameter
space (Berry and Wilkenson 1984)
7 Bounded Quantum Systems
γ = 10Y and β =
1 −
1
6
X
10Y + 10X,
where 0 ≤ X ≤ 9 and 0 ≤ Y ≤ 6. There are three cases for which this system
is known to be classically integrable, (α, β, γ ) = (60, 60, 60), (90, 45, 45), and
(90, 60, 30). For other cases, the system is classically chaotic. However, even when
the system is classically chaotic, there are special cases when spatial symmetries
exist. For example, when X = 9, the triangle is isosceles and has reflection
symmetry. In this case, the eigenstates will have either even or odd parity and the
spectrum is a mixed spectrum composed of two pure sequences, one due to even
parity states and the other due to odd parity states.
In Fig. 7.2b, we show the lowest 13 energy eigenstates for the case when X = 7
for the range 0.6 ≤ Y ≤ 6. For this range of angles, the system is classically
chaotic and no known symmetries exist except at Y = 6, where the system becomes
classically integrable. We see a number of avoided crossings in this figure but
no accidental degeneracies. At Y = 6, where the system is integrable, several
degeneracies occur. In Fig. 7.3a, we show the spectrum for the case X = 9. For this
value of X, the triangle is isosceles for all values of Y , and therefore, even though
the system is classically chaotic, a symmetry exists that causes the spectrum to form
a mixed sequence as discussed above. We see that a number of degeneracies occur
as we vary Y . These degeneracies are not accidental but due to the symmetry. Berry
and Wilkenson had to search hard for true accidental degeneracies. An example of
such a degeneracy is shown in Fig. 7.3b for X = 4.61. It occurs between levels 6
Fig. 7.3 Energy-level curves for the isosceles triangle. (a) X = 9: many level crossings occur
between states of different parity. The small boxes on the right indicate regions where more than
one degeneracy occurs. (b) X = 4.61: a true accidental degeneracy occurs between levels 6 and
7 at Y = 3.97. For this degeneracy, the wave function changes sign under a rotation in parameter
space (Berry and Wilkenson 1984)