An approach to establish a connection between algebraic and configuration spaces: su(ν + 1) algebraic model for vibrational excitations

Abstract

A realisation of coordinates and momenta in the algebraic space to describe vibrational excitations of ν-equivalent oscillators is obtained. The connection between algebraic and configuration spaces is carried out using the approach recently proposed [Mol. Phys. (2017), doi:10.1080/00268976.2017.1358829]. The realisation consists in an expansion in terms of the dynamical algebra generators with coefficients determined through a minimisation procedure and given in terms of matrix elements defined in configuration space. Two realisations are presented: one through an isomorphism with a harmonic oscillators basis and the second one using a mapping to Morse oscillators. In the case of the harmonic oscillator mapping, two chains associated with coordinate and momentum representation have been identified. Our approach allows us to establish the algebraic representation of any interaction and, consequently, of the Hamiltonian associated with ν-interacting oscillators, providing a formal approach to estimate the potential energy surface.

GRAPHICAL ABSTRACT

Keywords:

• algebraic approach
• configuration space
• algebraic realisation
• coordinates and momenta
• harmonic oscillators
• Morse oscillators

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Along this work, the usual convention of using upper-case roman letters for a Lie group and lower-case roman letters for the corresponding Lie algebra is used. In addition, it should be noted that special unitary and orthogonal groups/algebras appear when a scalar quantity in conserved due to the isomorphism . Thus, since this is our case, the use of or is equivalent.

2 Notice that to describe a system in equilibrium with a thermal bath of temperature T, we can use with .

Additional information

Funding

This work is partially supported by DEGAPA-UNAM (Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México), Mexico, under project IN-227017, by Spanish Consejería de Economía, Inovación, Ciencia y Empleo, Junta de Andalucía (Agencia de Innovación y Desarrollo de Andalucía) [grant numbers FQM-160, P11-FQM-7632] and by the Spanish Ministerio de Economía y Competitividad and FEDER funds under projects FIS2017-88410-P and FPA2016-77689-C2-1-R.