7.5 Quantum Billiards
217
Finally, in Fig. 7.16d, where a/d = 2/75, we have hard chaos. There are no island
structures remaining in the phase space (at least at the length scales we show in that
figure).
The ripple billiard has the special property that its energy eigenvalues and energy
eigenstates can be obtained from a Hamiltonian matrix (Luna-Acosta et al. 1996),
and matrix elements can be obtained analytically (Li et al. 2002). Consider a particle
of mass m = 1. In the (x, y) coordinate frame, the Schrödinger equation for the
particle inside the billiard is
−
1
2
∂ 2
∂x 2 +
∂ 2
∂y 2
ψ n (x, y) = E n ψ n (x, y),
(7.27)
where E n and ψ n (x, y) are the nth energy eigenvalue and eigenstate, respectively.
For a ripple billiard totally enclosed by hard walls, the boundary conditions are
ψ n (x, 0) = 0, ψ n (±π, y) = 0, ψ n (x, d + a cos(x)) = 0.
(7.28)
It is useful to change coordinates from the (x, y) frame to a new frame, (u, v), in
which the walls are straight. We let
u = x and v =
y
d + a ξ(x)
,
(7.29)
where ξ(x) = cos(x). In terms of the coordinates (u, v), the Schrödinger equation
takes the form
−
1
2
∂ 2
∂u 2 + h 1
∂ 2
∂v 2 + h 2
∂ 2
∂u∂v
+ h 3
∂
∂v
n (u, v) = E n n (u, v),
(7.30)
where n (u, v) = ψ(x = u, y = v/(d + aξ(u))),
h 1 =
1 + a 2 v 2 ξ 2
u
(d + aξ ) 2 , h 2 =
−2avξ u
(d + aξ )
, and h 3 =
−avξ uu
(d + aξ )
+
2a 2 vξ 2
u
(d + aξ ) 2 ,
(7.31)
with ξ u ≡
∂ξ
∂u . The boundary conditions in the (u, v) coordinate frame are
n (u, v = 0) = n (u, v = 1) = 0, and n (u = ±π, v) = 0.
(7.32)
The eigenfunctions, n (u, v), are normalized so that
π
−π
du
1
0
dv J (u, v) )
∗
n (u, v)) n (u, v) = 1,
(7.33)