232
7 Bounded Quantum Systems
as the second invariant operator. Then the commutator [ ˆ
T, ˆ
H ] =0, if the potential
energy is non-zero. If we let |E j denote energy eigenstates of ˆ
H on the unit cell,
we can write the time average of ˆ
T as
ˆ
T = lim τ →∞
1
τ
τ
0
dt ˆ
T(t) =
j
j | ˆ
T(0)|E j j j |.
(7.57)
In Fig. 7.27, we plot ˆ
T av = = ˆ
p 2
x + p 2
y versus E for two versions of the unit cell.
Fig. 7.27a shows a trivial plot of data points { ˆ
T av , E} because we have set U o = 0.
However, it does show the highly degenerate set of energy eigenvalues of a square
empty billiard. In Fig. 7.27b, we plot the data points { ˆ
T av , E} for the unit cell with
potential V (x, y). In Fig. 7.27b, the data points { ˆ
T av , E} are randomly scattered
about the straight line, indicating that the second constant of the motion (and second
good quantum number) has been destroyed. This behavior is consistent with what
we observe in the classical LSOS plot.
On an infinitely extended periodic lattice, Bloch momenta form a continuum
of values, but we can analyze the behavior of the lattice using periodic boundary
conditions for the Bloch states. For a square lattice composed of N 2 unit cells of the
infinitely extended lattice, the Bloch momenta would have values {k x =
x π
N , k y =
y π
N }, where −∞≤ x ≤∞ and −∞≤ y ≤∞. In the limit N→∞, the properties of
the two lattices become identical. We first discuss properties of the unit cell and then
look at the band structure of the extended lattice.
In Fig. 7.28, we show the Brillouin zone for the soft Sinai lattice. The solid
lines are symmetry lines and the points labeled X, and K are high symmetry
points in the Brillouin zone. Although the Brillouin zone extends over the range
of −
π
2 ≤k x ≤
π
2 , −
π
2 ≤k y ≤
π
2 , only a small fraction of the Bloch momenta in the
Fig. 7.27 Plots T av = = ˆ
p 2
x + ˆ
p 2
y versus E for two different lattices. (a) T x versus E for U o = 0.0.
(c) T x versus E for the lattice V (x, y) and U o = 120 (reproduced from Porter et al. 2017)
7 Bounded Quantum Systems
as the second invariant operator. Then the commutator [ ˆ
T, ˆ
H ] =0, if the potential
energy is non-zero. If we let |E j denote energy eigenstates of ˆ
H on the unit cell,
we can write the time average of ˆ
T as
ˆ
T = lim τ →∞
1
τ
τ
0
dt ˆ
T(t) =
j
j | ˆ
T(0)|E j j j |.
(7.57)
In Fig. 7.27, we plot ˆ
T av = = ˆ
p 2
x + p 2
y versus E for two versions of the unit cell.
Fig. 7.27a shows a trivial plot of data points { ˆ
T av , E} because we have set U o = 0.
However, it does show the highly degenerate set of energy eigenvalues of a square
empty billiard. In Fig. 7.27b, we plot the data points { ˆ
T av , E} for the unit cell with
potential V (x, y). In Fig. 7.27b, the data points { ˆ
T av , E} are randomly scattered
about the straight line, indicating that the second constant of the motion (and second
good quantum number) has been destroyed. This behavior is consistent with what
we observe in the classical LSOS plot.
On an infinitely extended periodic lattice, Bloch momenta form a continuum
of values, but we can analyze the behavior of the lattice using periodic boundary
conditions for the Bloch states. For a square lattice composed of N 2 unit cells of the
infinitely extended lattice, the Bloch momenta would have values {k x =
x π
N , k y =
y π
N }, where −∞≤ x ≤∞ and −∞≤ y ≤∞. In the limit N→∞, the properties of
the two lattices become identical. We first discuss properties of the unit cell and then
look at the band structure of the extended lattice.
In Fig. 7.28, we show the Brillouin zone for the soft Sinai lattice. The solid
lines are symmetry lines and the points labeled X, and K are high symmetry
points in the Brillouin zone. Although the Brillouin zone extends over the range
of −
π
2 ≤k x ≤
π
2 , −
π
2 ≤k y ≤
π
2 , only a small fraction of the Bloch momenta in the
Fig. 7.27 Plots T av = = ˆ
p 2
x + ˆ
p 2
y versus E for two different lattices. (a) T x versus E for U o = 0.0.
(c) T x versus E for the lattice V (x, y) and U o = 120 (reproduced from Porter et al. 2017)