SymbMat: symbolic computation of quantum transition matrix elements. We have developed a set of Mathematica notebooks to compute symbolically quantum transition matrices relevant for atomic ionization processes. The utilization of a symbolic language allows us to obtain analytical expressions for the transition matrix elements required in charged-particle and laser induced ionization of atoms. Additionally, by using a few simple commands, it is possible to export these symbolic expressions to standard programming languages, such as Fortran or C, for the subsequent computation of differential cross sections or other observables. One of the main drawbacks in the calculation of transition matrices is the tedious algebraic work required when initial states other than the simple hydrogenic 1$s$ state need to be considered. Using these notebooks the work is dramatically reduced and it is possible to generate $exact$ expressions for a large set of bound states. We present explicit examples of atomic collisions (in First Born Approximation and Distorted Wave Theory) and laser-matter interactions (within the Dipole and Strong Field Approximations and different gauges) using both hydrogenic wavefunctions and Slater-Type Orbitals with arbitrary $nlm$ quantum numbers as initial states.

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  1. Ciappina, M. F.; Kirchner, T.: S\textscymbM\textscat: symbolic computation of quantum transition matrix elements (2012)