124
8 Quantum Ratchets
to the Hamiltonian of Eq. 8.1 namely,
˙
x =
∂H
∂p
˙
p = −
∂H
∂x
(8.2)
˙
q k =
∂H
∂p k
˙
p k = −
∂H
∂q k
(8.3)
From Eqs. 8.1, 8.2, 8.3, we obtain,
¨
x + ω
2
0 x = −
g
m
k
β k q k
˙
x =
p
m
(8.4)
¨
q k + ω
2
k q k = −gβ k x
˙
q = p k
(8.5)
In accordance of Appendix, the solution of Eq. 8.5 is given by,
q k =
I m(ξ k
ω k
= −g
β k
ω k
t
0
x(τ )sin(ω k (t − τ ))dτ +
p k0
ω k
sin(ω k t) + q k0 cos(ω k t)
(8.6)
Replacing in the former Eq. 8.6 sin(ω k (t − τ ))dτ =
1
ω k
d[cos(ω k (t − τ ))], we get
q k =
I m(ξ k
ω k
= −g
β k
ω k
t
0
x(τ )d[cos(ω k (t − τ ))] +
p k0
ω k
sin(ω k t) + q k0 cos(ω k t)
(8.7)
Performing the integration by parts, we can write,
g
k
β k q k =
t
0
K (t − τ ) ˙
x(τ )dτ − K (0) x(t) + x(0)K (t) − f (t)
(8.8)
where
K (t) = g
2
k
β 2
ω 2
k
cos(ω k (t))
(8.9)
and
f (t) = −g
k
β k
q k0 cos(ω k (t)) +
p k0
ω k
sin(ω k (t))
(8.10)
and q k0 and p k0 are the initial values of the canonical variables. Replacing
g
k β k q k given by the former equation in Eq. 8.8, we obtain,