7.5 Quantum Billiards
219
which is a measure of the variation of the wave function along the horizontal
direction near the wall.
The Husimi distribution function (Husimi 1940), W (x 0 , p x0 ), of a state,
ψ n (x, y), can be viewed as a quantum mechanical phase space probability density
for that state. It is defined as
W n (x 0 , p x0 ) = ||ψ n |x 0 , p x0 |
2 ,
(7.39)
where the state |x 0 , p x0 is a minimum uncertainty wave packet that can be
represented in the position basis as
x|x 0 , p x0 =
ω
π
1/4
exp
−
ω
2
(x − x 0 )
2
+ ip x0 (x − x 0 )
.
(7.40)
The coherent state has a standard deviation
√
1/2ω in position and a standard
deviation
√
ω/2 in momentum (Louisell 1973). W (x 0 , p x0 ) gives the probability
to find the state |ψ n in a cell of size 1 (1 ¯
h in dimensioned units) centered at the
phase space point (p x0 , x 0 ).
The Husimi distribution of the eigenstate ψ n (x, y) near the boundary y ≈ 0 is
given by
W n (x 0 , p x0 ) =
dxx 0 , p x0 |xS n (x)
2
(7.41)
for hard wall boundary conditions. For periodic boundary conditions, we must use
a spatially periodic Husimi distribution function (Chang and Shi 1986),
x|x 0 , p x0 =
ω
π
1/4
∞
l=−∞
exp
−
ω
2
(x + 2πl − x 0 )
2
+ip x0 (x + 2πl − x 0 )
.
(7.42)
Husimi distributions for several of the energy eigenstates of the ripple billiard
are shown in Fig. 7.17 for the case a/d = 2/5. This case has the classical surface of
section shown in Fig. 7.16b. For each energy eigenstate, we chose the momentum
cutoff to be p E =
√
2E, where E is the energy of the eigenstate. In cases where
the probability extended beyond p E =
√
2E in momentum space, the plots were
extended to larger values of p x . The eigenstate in Fig. 7.17a is localized on a higherorder island chain. The eigenstates shown in Figs. 7.17b–d are spread throughout the
chaotic region and show evidence of scarring.
219
which is a measure of the variation of the wave function along the horizontal
direction near the wall.
The Husimi distribution function (Husimi 1940), W (x 0 , p x0 ), of a state,
ψ n (x, y), can be viewed as a quantum mechanical phase space probability density
for that state. It is defined as
W n (x 0 , p x0 ) = ||ψ n |x 0 , p x0 |
2 ,
(7.39)
where the state |x 0 , p x0 is a minimum uncertainty wave packet that can be
represented in the position basis as
x|x 0 , p x0 =
ω
π
1/4
exp
−
ω
2
(x − x 0 )
2
+ ip x0 (x − x 0 )
.
(7.40)
The coherent state has a standard deviation
√
1/2ω in position and a standard
deviation
√
ω/2 in momentum (Louisell 1973). W (x 0 , p x0 ) gives the probability
to find the state |ψ n in a cell of size 1 (1 ¯
h in dimensioned units) centered at the
phase space point (p x0 , x 0 ).
The Husimi distribution of the eigenstate ψ n (x, y) near the boundary y ≈ 0 is
given by
W n (x 0 , p x0 ) =
dxx 0 , p x0 |xS n (x)
2
(7.41)
for hard wall boundary conditions. For periodic boundary conditions, we must use
a spatially periodic Husimi distribution function (Chang and Shi 1986),
x|x 0 , p x0 =
ω
π
1/4
∞
l=−∞
exp
−
ω
2
(x + 2πl − x 0 )
2
+ip x0 (x + 2πl − x 0 )
.
(7.42)
Husimi distributions for several of the energy eigenstates of the ripple billiard
are shown in Fig. 7.17 for the case a/d = 2/5. This case has the classical surface of
section shown in Fig. 7.16b. For each energy eigenstate, we chose the momentum
cutoff to be p E =
√
2E, where E is the energy of the eigenstate. In cases where
the probability extended beyond p E =
√
2E in momentum space, the plots were
extended to larger values of p x . The eigenstate in Fig. 7.17a is localized on a higherorder island chain. The eigenstates shown in Figs. 7.17b–d are spread throughout the
chaotic region and show evidence of scarring.