4.1 Franck–Condon Excitation
121
one. From the above considerations about the relationship between ultrashort pulses
and the generation of well-localized excited wavepackets, we see that the shorter the
pulse, the closer to the real process is the vertical excitation assumption.
4.2 Vibrational Wavepacket Dynamics
When a wavepacket representing the nuclear motion is sufficiently well localized in
the nuclear phase space, it moves in a way that can be approximately described by
Newton-like equations. For objects with a large mass and momentum the indetermination on both position and momentum is physically irrelevant and Newton’s laws
of dynamics apply. Nuclei, especially lower down in the periodic table, are heavy
enough as to be treated by classical mechanics as a first approximation, although
important details, such as the quantization of vibrational levels and tunneling, are lost.
Ehrenfest’s theorem shows that there is a relationship between quantum wavepacket
dynamics and Newton’s laws even for light particles.
Consider the time dependence of the expectation value of an observable ˆ
A, for
a wavepacket ψ(x, t). Here x is the collection of the particles coordinates x i . From
the TDSE, Eq. (2.1), we have:
d
dt
ψ
ˆ
A
ψ
=
dψ
dt
ˆ
A
ψ
+
ψ
ˆ
A
dψ
dt
=
i
ψ
ˆ
H , ˆ
A
ψ
.
(4.4)
In the Hamiltonian ˆ
H , we must distinguish the potential energy term V (x) and the
kinetic energy
ˆ
T = −
j
2
2m j
∂
2
∂ x
2
j
.
(4.5)
Here m j is the mass of the particle associated with the x j coordinate. If ˆ
A is one of
the coordinates, say x i , it commutes with V (x) but not with ˆ
T . By indicating with
the shorthand x i its expectation value, we can write
d x i
dt
=
i
ψ
ˆ
T , x i
ψ
=
ˆ
p i
m i
(4.6)
where ˆ
p i is the linear momentum operator associated with the coordinate x i . Note
that only the derivative with respect to x i contributes to the commutator. Consider
now the time derivative of the expectation value of ˆ
p i . Here the only contribution to
the commutator is due to V (x):
d
ˆ
p i
dt
=
i
ψ
−i
V (x),
∂
∂ x i
ψ
= −
∂ V
∂ x i
.
(4.7)
121
one. From the above considerations about the relationship between ultrashort pulses
and the generation of well-localized excited wavepackets, we see that the shorter the
pulse, the closer to the real process is the vertical excitation assumption.
4.2 Vibrational Wavepacket Dynamics
When a wavepacket representing the nuclear motion is sufficiently well localized in
the nuclear phase space, it moves in a way that can be approximately described by
Newton-like equations. For objects with a large mass and momentum the indetermination on both position and momentum is physically irrelevant and Newton’s laws
of dynamics apply. Nuclei, especially lower down in the periodic table, are heavy
enough as to be treated by classical mechanics as a first approximation, although
important details, such as the quantization of vibrational levels and tunneling, are lost.
Ehrenfest’s theorem shows that there is a relationship between quantum wavepacket
dynamics and Newton’s laws even for light particles.
Consider the time dependence of the expectation value of an observable ˆ
A, for
a wavepacket ψ(x, t). Here x is the collection of the particles coordinates x i . From
the TDSE, Eq. (2.1), we have:
d
dt
ψ
ˆ
A
ψ
=
dψ
dt
ˆ
A
ψ
+
ψ
ˆ
A
dψ
dt
=
i
ψ
ˆ
H , ˆ
A
ψ
.
(4.4)
In the Hamiltonian ˆ
H , we must distinguish the potential energy term V (x) and the
kinetic energy
ˆ
T = −
j
2
2m j
∂
2
∂ x
2
j
.
(4.5)
Here m j is the mass of the particle associated with the x j coordinate. If ˆ
A is one of
the coordinates, say x i , it commutes with V (x) but not with ˆ
T . By indicating with
the shorthand x i its expectation value, we can write
d x i
dt
=
i
ψ
ˆ
T , x i
ψ
=
ˆ
p i
m i
(4.6)
where ˆ
p i is the linear momentum operator associated with the coordinate x i . Note
that only the derivative with respect to x i contributes to the commutator. Consider
now the time derivative of the expectation value of ˆ
p i . Here the only contribution to
the commutator is due to V (x):
d
ˆ
p i
dt
=
i
ψ
−i
V (x),
∂
∂ x i
ψ
= −
∂ V
∂ x i
.
(4.7)