Modeling highly symmetric quantum systems with regularized [delta]-function potentials in two and three dimensions
The regularized [delta]-function potential model is used to compute optical and electronic properties of highly symmetric quantum systems. Dirac [delta]-functions allow us to very easily simulate complex quantum systems. In particular, when investigating effects due to symmetric properties of the system, other functional models -- like, for example, elliptic-[theta] functions -- prove to be more computationally expensive than necessary. In this dissertation, we demonstrate the use of regularized [delta]-function potential models in the simulation of a single-walled carbon nanotube and a tetrahedral molecule. Specifically, we show that, when a nanotube with armchair chirality is driven by a circularly polarized optical field, it generates selective high-order harmonic radiation. Using Floquet-Bloch theory, we compute the quasienergy and average energy band structure, as well as the non-linear electron current and emitted high-harmonic power spectrum. Furthermore, we compute and visualize the quasibound state wavefunctions of electrons in a tetrahedrally symmetric molecule. Quasibound states live in the positive energy continuum of molecules, and, due to their relatively long lifetimes, they can have a great impact on dynamics of chemical reactions. Using Wigner-Eisenbud theory and a basis of tetrahedral harmonic functions, we compute the electron scattering dynamics of such a system and show correspondence between the quasibound state energies and the complex poles of the S-matrix.