Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

We give a brief and basic introduction to perturbation theory. The main idea is to attempt to consider the situation of interest as a small perturbation of a simpler situation (which can be understood completely), and in particular to consider a system with a weak interaction as a perturbation of a non-interacting system. We develop the interaction picture, which allows approximating the time-evolution of an interacting system by the partial sums of the Dyson series, a fundamental tool for the sequel. We illustrate these ideas on a rudimentary model of the interaction of electrons and “photons”.

The fundamentals are introduced, starting with the interaction of one system with another at the quantum level, and introducing the perturbation series. In the case of a gas interacting with a weak electromagnetic field, the series may be truncated at the linear term and used to provide the rate at which radiatively induced dipole transitions will occur between any two spectral states of the gas. Under equilibrium conditions this rate will exhibit a balance between the induced emission and pure absorption that is characterized by the important fluctuation–dissipation theorem, and leads, when translated into a power loss in the field, to an expression for the net absorption coefficient. Within this, there resides a spectral density function, essentially the line shape, which is shown to be the Fourier transform of the autocorrelation function for the active dipole moment, a statistical measure for the fluctuations in the dipole due to molecular collisions within the gas. The spectral density may then be identified, through the Wiener–Khinchin theorem, with the power spectrum of the dipole fluctuations, and stands ready for a reduction to the molecular level.

A sample of gas, originally treated as a single quantum system, is now described in terms of its molecular constituents, starting with the case of a single radiating molecule in an equilibrium bath of perturbers. First, the isolated radiator is considered, as if the bath had been deactivated, allowing a discussion of how its internal energy and angular momentum may change when, in the presence of an electromagnetic field , a radiant transition takes place, and of how the transition amplitude may be reduced under the Wigner–Eckart theorem. Then, the interaction between radiator and bath is reinstated, but the initial correlations between the two are neglected, so that a separate average over the bath may be taken. There is then an examination of various approximations that may be of use elsewhere. These are the restrictions to collisions that are binary in nature, the possibility that a collision may be said to follow a classical trajectory, and the validity of treating it under the impact approximation, which carries a restriction to the core region of a spectral line, but offers a great simplification when collisions may be regarded as very brief, well-separated events.