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Journal of Mathematical Physics

Factorization−method treatment of the perturbed Morse oscillator

J. N. Huffaker, Parmesh H. Dwivedi

Published Apr 1, 1975• 77 citations
Publisher

Abstract

By transforming the Morse oscillator from type B to type F factorization, we obtain operators for raising and lowering the vibrational quantum number v, and a recursion relation. This permits the calculation of matrix elements needed for perturbation of the Morse oscillator to any order for perturbations of the form (eau − 1)n. Explicit calculations are presented for n = 3 and n = 4, analogous to the usual cubic and quartic anharmonicity perturbations of the harmonic oscillator.

DOI
10.1063/1.522633

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By transforming the Morse oscillator from type B to type F factorization, we obtain operators for raising and lowering the vibrational quantum number v, and a recursion relation. This permits the calculation of matrix elements needed for perturbation of the Morse oscillator to any order for perturbations of the form (eau − 1)n. Explicit calculations are presented for n = 3 and n = 4, analogous to the usual cubic and quartic anharmonicity perturbations of the harmonic oscillator.
A method for calculating the analytical solutions of the one-dimensional Schr\"odinger equation is suggested. A general discussion of the possible forms of the potentials and wave functions that are necessary to get the analytical solution is presented. In general, the analytical solutions appear in multiplets corresponding to the quantum number n of the harmonic oscillator. As an application, known solutions for the anharmonic oscillators are critically recalculated and a few additional results are found. Analytical solutions are also found for the generalized Morse oscillators. \textcopyright{} 1996 The American Physical Society.
Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
A realization of the raising and lowering operators for the Morse potential is presented. We show that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are derived for the matrix elements of different functions such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied. The transition probability between two eigenstates produced by a harmonic perturbation as a function of the operators [Formula: see text]±,0 is discussed. The average values of some observables in the coherent states |α > for the Morse potential are also calculated. PACS Nos.: 02.30+b, 03.65Fd, 42.50Ar, and 33.10Cs