The time evolution of a single electronic wave function in an atom,  is described by the Schrödinger equation (in atomic units):
where the light-matter interaction Hamiltonian, , can be expressed in the length gauge, within the dipole approximation, as:
where  is the Coulomb potential of the atomic species considered;  are the momentum and position operator, respectively; and  is the total electric field evaluated in the neighbor of the atom.
The formal solution of the Schrödinger equation is given by the propagator formalism:
where , is the electron wave function at time .
This exact solution cannot be used for almost any practical purpose.
However, it can be proved, using Dyson's equations that the previous solution can also be written as:
where,
is the bounded Hamiltonian and 
is the interaction Hamiltonian.
The formal solution of Eq. , which previously was simply written as Eq. , can now be regarded in Eq.  as a superposition of different quantum paths (or quantum trajectory), each one of them with a peculiar interaction time  with the electric field.
In other words, each quantum path is characterized by three steps:
# An initial evolution without the electromagnetic field. This is described by the left-hand side  term in the integral.
# Then, a "kick" from the electromagnetic field,  that "excite" the electron. This event occurs at an arbitrary time that uni-vocally characterizes the quantum path .
#A final evolution driven by both the field and the Coulomb potential, given by .
In parallel, you also have a quantum path that do not perceive the field at all, this trajectory is indicated by the right-hand side term in Eq. .
This process is entirely time-reversible, i.e. can also occur in the opposite order.
Equation  is not straightforward to handle. However, physicists use it as the starting point for numerical calculation, more advanced discussion or several approximations.
For strong-field interaction problems, where ionization may occur, one can imagine to project Eq.  in a certain continuum state (unbounded state or free state) , of momentum , so that:
where <math>|c_{\textbf{p}}(t)|^2
, the electron in the continuum states .
If this probability amplitude is greater than zero, the electron is photoionized.
For the majority of application, the second term in  is not considered, and only the first one is used in discussions, hence:
Equation  is also known as time reversed S-matrix amplitude and it gives the probability of photoionization by a generic time-varying electric field.