4.2 Vibrational Wavepacket Dynamics
127
These equations too have classical analogues. Consider a set of classical trajectories
with different initial conditions for a system of particles, that give place to a distribution of values for the variables x i and p i . We define averages over all trajectories
for these variables and their second moments, and we indicate them as x i , p i , and so
on, to distinguish them from the quantum expectation values. Equation (4.8) trivially
yield the time dependence of x and p i :
d
dt
x i =
p i
m i
and
d
dt
p i = −
∂ V
∂ x i
.
(4.18)
For the second moments we get
d
dt
(x i − x i )
2
= 2(x i − x i )
˙
x i − ˙
x i
= 2m
−1
i (x i − x i ) (p i − p i )
(4.19)
d
dt
( p i − p i )
2
= −2( p i − p i )
∂ V
∂ x i
−
∂ V
∂ x i
.
(4.20)
We see that Eqs. (4.19) and (4.20) are perfectly analogue to Eqs. (4.16) and (4.17),
except that in the quantum case ˆ
p i does not commute with x i and V (x), so we have
the sum of two distinct quantities such as x i ˆ
p i and ˆ
p i x i instead of the classical
expression 2x i p i . The physical meaning of both pairs of equations concerns the
correlation between positions, momenta, and forces, which is more easily understood
in classical terms. According to Eq. (4.19) the x i distribution broadens in time if
x i − x i and p i − p i are mostly of the same sign, whereas it narrows if x i and p i
tend to deviate from their averages in opposite directions. Equation (4.20) similarly
means that the p i distribution spreads out when the momentum p i and the force
−∂ V /∂ x i deviate from their averages in a concordant way, while it narrows when
the deviations are discordant.
The consequences of such correlations for quantum dynamics are most clearly
illustrated by the Animations 4.1 and 4.4, which show the motion of well-localized
wavepackets created by Franck–Condon excitation. While the wavepacket oscillates
back and forth, its width in the R coordinate also changes. The width variation of
course is perfectly periodic in the harmonic potential and only approximately so in the
Morse one. The ground-state v = 0 eigenfunction that is translated into the excited
PES by an ultrashort pulse is a minimum uncertainty Gaussian wavepacket with
vanishing average momentum. Its R and ˆ
P (the associate momentum) distributions
are independent, so the expression Eq. 4.16 vanishes. However, if put in a constant
potential, the wavepacket would spread out indefinitely in the R coordinate, while
its momentum distribution would remain unchanged. Also in a linear potential, with
constant ∂ V /∂ R, the wavepacket would spread, but in this case
ˆ
P
would change in
time according to (4.7), still keeping the same distribution of ˆ
P −
ˆ
P
. We easily see
why a swarm of trajectories with the same distributions of coordinates and momenta
127
These equations too have classical analogues. Consider a set of classical trajectories
with different initial conditions for a system of particles, that give place to a distribution of values for the variables x i and p i . We define averages over all trajectories
for these variables and their second moments, and we indicate them as x i , p i , and so
on, to distinguish them from the quantum expectation values. Equation (4.8) trivially
yield the time dependence of x and p i :
d
dt
x i =
p i
m i
and
d
dt
p i = −
∂ V
∂ x i
.
(4.18)
For the second moments we get
d
dt
(x i − x i )
2
= 2(x i − x i )
˙
x i − ˙
x i
= 2m
−1
i (x i − x i ) (p i − p i )
(4.19)
d
dt
( p i − p i )
2
= −2( p i − p i )
∂ V
∂ x i
−
∂ V
∂ x i
.
(4.20)
We see that Eqs. (4.19) and (4.20) are perfectly analogue to Eqs. (4.16) and (4.17),
except that in the quantum case ˆ
p i does not commute with x i and V (x), so we have
the sum of two distinct quantities such as x i ˆ
p i and ˆ
p i x i instead of the classical
expression 2x i p i . The physical meaning of both pairs of equations concerns the
correlation between positions, momenta, and forces, which is more easily understood
in classical terms. According to Eq. (4.19) the x i distribution broadens in time if
x i − x i and p i − p i are mostly of the same sign, whereas it narrows if x i and p i
tend to deviate from their averages in opposite directions. Equation (4.20) similarly
means that the p i distribution spreads out when the momentum p i and the force
−∂ V /∂ x i deviate from their averages in a concordant way, while it narrows when
the deviations are discordant.
The consequences of such correlations for quantum dynamics are most clearly
illustrated by the Animations 4.1 and 4.4, which show the motion of well-localized
wavepackets created by Franck–Condon excitation. While the wavepacket oscillates
back and forth, its width in the R coordinate also changes. The width variation of
course is perfectly periodic in the harmonic potential and only approximately so in the
Morse one. The ground-state v = 0 eigenfunction that is translated into the excited
PES by an ultrashort pulse is a minimum uncertainty Gaussian wavepacket with
vanishing average momentum. Its R and ˆ
P (the associate momentum) distributions
are independent, so the expression Eq. 4.16 vanishes. However, if put in a constant
potential, the wavepacket would spread out indefinitely in the R coordinate, while
its momentum distribution would remain unchanged. Also in a linear potential, with
constant ∂ V /∂ R, the wavepacket would spread, but in this case
ˆ
P
would change in
time according to (4.7), still keeping the same distribution of ˆ
P −
ˆ
P
. We easily see
why a swarm of trajectories with the same distributions of coordinates and momenta