Time-dependent ultrafast diffraction measurements can be directly
inverted to obtain the dynamics of atomic
motions, in contrast to ultrafast spectra which require detailed
knowledge of the sample (e.g., potential energy
surfaces) for their inversion. We consider here how to derive
time-dependent diffraction (the X-ray and
electron diffraction cases being very similar) from nuclear quantum
dynamics and vice versa and how this
may be used to directly observe the atomic motions in molecules, in
particular how chemical reactions take
place. Two simple examples of dissociative and bound quantum
(vibrational and rotational) dynamics in a
gas-phase sample of diatomic molecules, excited by an optical pump
pulse and measured by an electron or
X-ray probe pulse, are presented. The quantum mechanical basis of
the breaking of symmetry due to the
linearly polarized optical pump pulse and the superposition and
interference between the ground and excited
electronic states are discussed. We demonstrate how to isolate the
short-time excited-state dynamics from
that of the ground state using the symmetry of the electronic dipole
transition. We illustrate that the time-evolving distribution of interatomic distances can be clearly resolved
from the ultrafast diffraction data and
thus illustrate that the detailed dynamics of molecular vibration and
the progress of a photodissociation reaction
could be watched as they occur. In addition, we show that the
duration of ultrafast X-ray and electron pulses
can be measured with a time resolution of tens of femtoseconds by
clocking it against such atomic motion.