Ladder operators for Morse oscillator and a perturbed vibrational problem

ABSTRACT

Quantum-mechanical methods of solving the polyatomic vibrational Schrödinger equation need higher quality zero-order approximations than ones originating from the harmonic oscillator (HO). Ladder operators built on the HO have a number of unique features simplifying both the operator perturbation theory and practical implementations of matrix-elements-based methods. Therefore, finding suitable ladder operators for solvable anharmonic oscillators and mainly the Morse oscillator remain one of the major challenges of nuclear vibrational dynamics. In this work, we review the problem of building Morse oscillator ladder operators (MLOs) and the prospects of their use in various methods of solving the many-dimensional anharmonic vibrational problem. The features of several existing approaches for building MLOs are explored and analysed. The native MLOs obtained by the factorisation method are not quite suitable for expressing a perturbed potential energy operator. Supersymmetric quantum mechanics (SUSYQM) does not solve the problem either since corresponding ladder operators only connect states from related potentials. The SU(2) vibron model provides an approximate solution based on a formal isomorphism of energy states. We have found that for the present the only useful model for MLOs is based on the so-called quasi-number states basis set (QNSB) built on modified Laguerre polynomials. QNSB yields a finite tridiagonal matrix representation of the Morse Hamiltonian corresponding to the exact solution. The convenience and accuracy of QNSB approach in comparison to second/fourth-order perturbation theory is illustrated with the HF molecule. The general conclusion is that QNSB-based MLOs are suitable for building many-body treatments, for instance, with the VSCF/VCI approach.

KEYWORDS:

• Morse oscillator
• ladder operators
• anharmonic vibrations
• polyatomic molecule
• perturbation theory
• quasi-number states

Acknowledgement

The authors are indebted to Prof. Norman C. Craig from Oberlin College (Oberlin/OH, USA) for reading the manuscript and helpful suggestions; to Vladimir I. Pupyshev from Lomonosov Moscow State University (Moscow, Russian Federation) for many valuable discussions; and to Prof. Oleg N. Ulenekov and Prof. Elena S. Bekhtereva from Tomsk University (Tomsk, Russian Federation) for general support. We would like to thank the reviewer for the detailed comments and suggestions for the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Sergey V. Krasnoshchekov http://orcid.org/0000-0003-2562-500X