Unitary group approach to describe interatomic potentials in 3D systems

Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

Algebraic discrete variable representation (DVR) methods that have been recently proposed are applied to describe 1D and 2D piecewise potentials. First, it is shown that it is possible to use a DVR approach to describe 1D square well potentials testing the wave functions with exact results. Thereafter, Morse and Pöschl-Teller (PT) potentials are described with multistep piecewise potentials in order to explore the sensibility of the potential to reproduce the transition from a pure square well energy pattern to an anharmonic energy spectrum. Once the properties of the different algebraic DVR approaches are identified, the 2D square potential as a function of the potential depth is studied. We show that this system displays natural degeneracy, accidental degeneracy and systematic accidental degeneracy. The latter appears only for a confined potential, where the symmetry group is identified and irreducible representations are constructed. One particle confined in a rectangular well potential with commensurate sides is also analyzed. It is proved that the systematic accidental degeneracy appearing in this system is removed for finite potential depth.

Algebraic DVR Approaches Applied to Describe the Stark Effect

Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0.