  We discuss a method to follow step-by-step time evolution of atomic and
molecular systems based on QED (Quantum Electrodynamics). Our strategy includes
expanding the electron field operator by localized wavepackets to define
creation and annihilation operators and following the time evolution using the
equations of motion of the field operator in the Heisenberg picture. We first
derive a time evolution equation for the excitation operator, the product of
two creation or annihilation operators, which is necessary for constructing
operators of physical quantities such as the electronic charge density
operator. We then describe our approximation methods to obtain time
differential equations of the electronic density matrix, which is defined as
the expectation value of the excitation operator. By solving the equations
numerically, we show "electron-positron oscillations", the fluctuations
originated from virtual electron-positron pair creations and annihilations,
appear in the charge density of a hydrogen atom and molecule. We also show that
the period of the electron-positron oscillations becomes shorter by including
the self-energy process, in which the electron emits a photon and then absorbs
it again, and it can be interpreted as the increase in the electron mass due to
the self-energy.
