Algebraic DVR Approaches Applied to Describe the Stark Effect

Marisol Bermúdez-Montaña,José M Arias,Joaquín Gómez-Camacho,Emilio Orgaz,Renato Lemus,Marisol Rodríguez-Arcos

doi:10.3390/sym12101719

Marisol Bermúdez-Montaña, José M Arias + Show 4 more

Open Access

https://doi.org/10.3390/sym12101719

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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.

Highlights

A discrete variable representation approach is based on the search of a discrete basis in terms of which any function of the coordinates is diagonal
The discrete variable representation methods (DVR) in configuration space were developed with some variants since the 1960s, but systematically widely used during the 1980s with different names: discrete-variable representation method [1,2,3,4], quadrature discretization method [5,6], configuration localized states (CLS) approach [7], and Lagrange mesh method (LMM) [8,9,10,11], whose similarities and differences are discussed in Ref. [11]
DVR method consists of taking a cutoff 3D harmonic oscillator in Fock space and diagonalize the matrix representation of the squares of the coordinates and momenta to obtain two discrete bases

Summary

Introduction

A discrete variable representation approach is based on the search of a discrete basis in terms of which any function of the coordinates is diagonal. We introduce two algebraic DVR methods where the discrete bases are obtained using purely algebraic means without any explicit reference to polynomials This is accomplished through the diagonalization of the matrix representations associated with the coordinates and momenta. DVR method consists of taking a cutoff 3D harmonic oscillator in Fock space and diagonalize the matrix representation of the squares of the coordinates and momenta to obtain two discrete bases (HO-DVR approach), which in turn are used to diagonalize the matrix representation of the Hamiltonian in a simple manner Both methods are based on a harmonic oscillator basis, albeit they provide different convergence behavior due to the fact that the provided discrete bases are distinct.

Algebraic DVR Methods in 1D Systems

HO-DVR Approach

Coulombic Potential

Stark Effect

Conclusions