Algebraic DVR Approaches Applied to Describe the Stark Effect

Abstract

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.

Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

“…In order to show that the energy convergence showed in Table 3 provides also the correct wave functions, we have projected the eigenvectors to the position representation. Each eigenvector |ψ Γγ α of Hamiltonian ( 51) is given in terms of a linear combination of the symmetry adapted basis (50).…”

“…Both the U(n + 1)-UGA and HO-DVR methods are defined in terms of the harmonic oscillator basis, but they provide different discrete representation bases and consequently one method may be more suitable for a specific problem [49]. In fact, in the description of the Stark effect in the non-relativistic Hydrogen atom, U(4)-UGA turns out to be more suitable for describing the breaking of spherical symmetry due to the electric field [50], while in all tested potentials where the angular momentum is conserved, HO-DVR offers better convergence [49].…”

Symmetry Analysis of the Square Well Potential

“…The resulting discrete basis provides a DVR basis which corresponds to the zeros of the polynomials associated with the solutions of the harmonic oscillator. Both the U (n + 1)-UGA and HO-DVR methods takes an harmonic oscillator basis but they provide different results [47,48].…”

Study of the Lennard-Jones and H2 potentials in the framework of the Morse based algebraic DVR approach

“…It is possible to obtain a diagonal representation of a function of the coordinate, in particular the potential, in terms of this discrete basis. An algebraic DVR approach was proposed through the representation of operators in configuration space [27], one of the first attempts was the U (n + 1)-unitary group approach (UGA) in which the discrete bases are provided by the different dynamical symmetries associated with the U (n+1) dynamical group [56][57][58][59][60][61]. The generalization of the algebraic DVR methods is described in Ref.…”