122
4 Wavepacket Dynamics and Geometrical Relaxation
Ehrenfest’s equations (4.6) and (4.7) are the quantum equivalent of Newton’s equations, i.e.,
˙
x i =
p i
m i
and
˙
p i = −
∂ V
∂ x i
.
(4.8)
The difference is that Ehrenfest’s equations of course apply to the expectation values
of x i , ˆ
p i and ∂ V /∂ x i , i.e., to averages taken over all coordinates, while Newton’s
equations concern a single point in the phase space. Knowing x i and
ˆ
p i
is a
very valuable information if the coordinate and momentum distributions are sharply
peaked around the average values, but of course the indetermination principle limits
the localization of a wavepacket in both variables.
The shape of the potential energy function is also important. Let’s expand V (x)
as a Taylor series of Δx = x − x:
V (x) = V (x) + Δx
t G(x) +
1
2
Δx
t H(x)Δx + O
|Δx|
3
(4.9)
where G is the gradient of V (x) and H its Hessian matrix. Then
∂ V
∂ x i
= G i (x) +
j
H i j (x)Δx j + O
|Δx|
2
(4.10)
When we take the average of this distribution the linear term containing the Hessian
does not contribute, so by neglecting the terms of the order of |Δx|
3 in the potential
we get
∂ V
∂ x i
∂ V
∂ x i
x
.
(4.11)
Then, the time evolution of x and p does not depend on the average of G, but
just on its value in x. This result is even closer to classical mechanics than Eq. (4.7)
and is valid as far as the wavepacket is sufficiently localized in space; otherwise one
cannot neglect the higher powers of |Δx|.
For harmonic potentials Eq. (4.11) is exact, so the evolution of x and p is
perfectly classical. By converting to the normal coordinates system (see Sect. 2.5), the
motion is periodic along each coordinate Q α . The periodicity of quantum dynamics
is a more general property of the harmonic oscillator and is due to the equal spacing of
the energy levels E v = (v + 1/2)ω. The time-dependent wavefunction for a given
mode Q can be expressed as
χ(Q, t) =
∞
v=0
c v e
−i(v+1/2)ωt
χ v (Q) = e
−iωt/2
∞
v=0
c v e
−ivωt
χ v (Q)
(4.12)
where the χ v are the vibrational eigenfunctions. All the exp(−ivωt) factors are periodic functions of time, with a common period T = 2π/ω. So, apart from the irrelevant
4 Wavepacket Dynamics and Geometrical Relaxation
Ehrenfest’s equations (4.6) and (4.7) are the quantum equivalent of Newton’s equations, i.e.,
˙
x i =
p i
m i
and
˙
p i = −
∂ V
∂ x i
.
(4.8)
The difference is that Ehrenfest’s equations of course apply to the expectation values
of x i , ˆ
p i and ∂ V /∂ x i , i.e., to averages taken over all coordinates, while Newton’s
equations concern a single point in the phase space. Knowing x i and
ˆ
p i
is a
very valuable information if the coordinate and momentum distributions are sharply
peaked around the average values, but of course the indetermination principle limits
the localization of a wavepacket in both variables.
The shape of the potential energy function is also important. Let’s expand V (x)
as a Taylor series of Δx = x − x:
V (x) = V (x) + Δx
t G(x) +
1
2
Δx
t H(x)Δx + O
|Δx|
3
(4.9)
where G is the gradient of V (x) and H its Hessian matrix. Then
∂ V
∂ x i
= G i (x) +
j
H i j (x)Δx j + O
|Δx|
2
(4.10)
When we take the average of this distribution the linear term containing the Hessian
does not contribute, so by neglecting the terms of the order of |Δx|
3 in the potential
we get
∂ V
∂ x i
∂ V
∂ x i
x
.
(4.11)
Then, the time evolution of x and p does not depend on the average of G, but
just on its value in x. This result is even closer to classical mechanics than Eq. (4.7)
and is valid as far as the wavepacket is sufficiently localized in space; otherwise one
cannot neglect the higher powers of |Δx|.
For harmonic potentials Eq. (4.11) is exact, so the evolution of x and p is
perfectly classical. By converting to the normal coordinates system (see Sect. 2.5), the
motion is periodic along each coordinate Q α . The periodicity of quantum dynamics
is a more general property of the harmonic oscillator and is due to the equal spacing of
the energy levels E v = (v + 1/2)ω. The time-dependent wavefunction for a given
mode Q can be expressed as
χ(Q, t) =
∞
v=0
c v e
−i(v+1/2)ωt
χ v (Q) = e
−iωt/2
∞
v=0
c v e
−ivωt
χ v (Q)
(4.12)
where the χ v are the vibrational eigenfunctions. All the exp(−ivωt) factors are periodic functions of time, with a common period T = 2π/ω. So, apart from the irrelevant