A simple approach to solve the time independent Schröedinger equation for 1D systems

Lemus, R.

doi:10.1088/2399-6528/ab0617

Cited by 10 publications

(8 citation statements)

References 29 publications

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“…The 1D unitary group approach has already been established by Lemus [43,44]. Here, we present salient features of the method.…”

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

“…associated with the coordinates, the Hamiltonian in the energy representation takes the following form [43,44].…”

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

“…A remarkable fact of this approach is that the kets |[N]n are identified with the 1D harmonic oscillator, and consequently, it is possible to project kets |[N]ζ and |[N] ζ to position representations. It is important to take into account that, in the UGA, an accidental degeneracy appears, which must be removed [43,44]. This goal is achieved by introducing parameter in the matrix representation of the deformed oscillator contribution:…”

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

“…In this scheme, it is through the transformation brackets connecting the different bases that makes it possible to obtain the representation of the Hamiltonian matrix in a simple form in terms of diagonal matrices, a feature characterizing a DVR approach. This approach has been called the unitary group approach (UGA), and it has been presented in detail for nD potentials [43][44][45][46][47][48]. A natural alternative to this approach consists in projecting the harmonic oscillator basis to a subspace of finite dimension.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

2022

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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.

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“…The 1D unitary group approach has already been established by Lemus [43,44]. Here, we present salient features of the method.…”

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

“…associated with the coordinates, the Hamiltonian in the energy representation takes the following form [43,44].…”

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

Section: Informed Consent Statement: Not Applicablementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

2022

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“…This is accomplished through the diagonalization of the matrix representations associated with the coordinates and momenta. From this perspective, the recently-proposed unitary group approach (U(4)-UGA) belongs to an algebraic DVR method [12][13][14][15][16]. In this approach, the discretization provided by the zeros of the polynomials in configuration space is substituted by the branching rules involved in the subgroup chains embedded in the dynamical group.…”

Section: Introductionmentioning

confidence: 99%

Algebraic DVR Approaches Applied to Describe the Stark Effect

et al. 2020

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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.

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An algebraic approach to calculate Franck–Condon factors

Lemus

2019

J Math Chem

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An algebraic approach based on unitary algebras to calculate Franck-Condon factors (FCF's) is presented. The method is based on the unitary group approach, which consists in adding a scalar boson to the ν-D harmonic oscillator space, taking advantage of the transformation brackets connecting the energy, coordinate and momentum representations. In this scheme the solutions are given in terms of an expansion of harmonic oscillator functions. In this way the overlaps are expressed in terms of simple scalar product of the eigenvectors. As a benchmark to illustrate our approach the case of two 1D-Morse potentials is presented. Discussions concerned with both non-Condon contributions and the harmonic limit are included. FCF's involving the 1D-Morse and asymmetric double Morse potentials are also considered. As an application, the FCF's involving the S-S stretching in the S 2 O molecule are described in terms of two Morse potentials.

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A simple approach to solve the time independent Schröedinger equation for 1D systems

Cited by 10 publications

References 29 publications

Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy

Algebraic DVR Approaches Applied to Describe the Stark Effect

An algebraic approach to calculate Franck–Condon factors

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