Gauge invariance was discovered in the development of classical electromagnetism and was required when the latter was formulated in terms of the scalar and vector potentials. It is now considered to be a fundamental principle of nature, stating that
different forms of these potentials yield the same physical description: they describe the same electromagnetic field as long as they are related to each other by gauge transformations. Gauge invariance can also be included into the quantum description of matter interacting with an electromagnetic field by assuming that the wave function transforms under a given local unitary transformation. The result of this procedure is a quantum theory describing the coupling of electrons, nuclei and photons. Therefore, it is a very important concept: it is used in almost every fields of physics and it has been generalized to describe electroweak and strong interactions in the standard model of particles. A review of quantum mechanical gauge invariance and general unitary transformations is presented for atoms and molecules in interaction with intense short laser pulses, spanning the perturbative to highly nonlinear nonperturbative interaction regimes. Various unitary transformations for single spinless particle Time Dependent Schrodinger Equations, TDSE, are shown to correspond to different time-dependent Hamiltonians and wave functions. Accuracy of approximation methods involved in solutions of TDSEs such as perturbation theory and popular numerical methods depend on gauge or representation choices which can be more convenient due to faster convergence criteria. We focus on three main representations: length and velocity gauges, in addition to the acceleration form which is not a gauge, to describe perturbative and nonperturbative radiative interactions. Numerical schemes for solving TDSEs in different representations are also discussed.
A simple 1-D relativistic model for a diatomic molecule with a double point interaction potential is solved exactly in a constant electric field. The Weyl-Titchmarsh-Kodaira method is used to evaluate the spectral density function, allowing the corre
ct normalization of continuum states. The boundary conditions at the potential wells are evaluated using Colombeaus generalized function theory along with charge conjugation invariance and general properties of self-adjoint extensions for point-like interactions. The resulting spectral density function exhibits resonances for quasibound states which move in the complex energy plane as the model parameters are varied. It is observed that for a monotonically increasing interatomic distance, the ground state resonance can either go deeper into the negative continuum or can give rise to a sequence of avoided crossings, depending on the strength of the potential wells. For sufficiently low electric field strength or small interatomic distance, the behavior of resonances is qualitatively similar to non-relativistic results.
The validation and parallel implementation of a numerical method for the solution of the time-dependent Dirac equation is presented. This numerical method is based on a split operator scheme where the space-time dependence is computed in coordinate s
pace using the method of characteristics. Thus, most of the steps in the splitting are calculated exactly, making for a very efficient and unconditionally stable method. We show that it is free from spurious solutions related to the fermion-doubling problem and that it can be parallelized very efficiently. We consider a few simple physical systems such as the time evolution of Gaussian wave packets and the Klein paradox. The numerical results obtained are compared to analytical formulas for the validation of the method.
