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The Journal of Chemical Physics

Quantum Dynamics of Anharmonic Oscillators. III. The Morse Oscillator

Paul F. Endres, David J. Wilson

Published Jan 15, 1967• 26 citations
Publisher

Abstract

Quantum calculations of the dynamics of a Morse oscillator with a Gaussian initial wave packet have been performed. As previously observed in other simple oscillators the classical and quantum expectation values for the coordinate exhibit significantly different behaviors, especially after the first few cycles. Some theories of chemical kinetics require the classical-mechanical calculation of lifetimes which may be quite long compared to vibrational periods. These theories deserve re-examination in view of the dynamical behavior observed in the present work.

DOI
10.1063/1.1840683

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Quantum calculations of the dynamics of a Morse oscillator with a Gaussian initial wave packet have been performed. As previously observed in other simple oscillators the classical and quantum expectation values for the coordinate exhibit significantly different behaviors, especially after the first few cycles. Some theories of chemical kinetics require the classical-mechanical calculation of lifetimes which may be quite long compared to vibrational periods. These theories deserve re-examination in view of the dynamical behavior observed in the present work.
The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.
We study quantum mechanical behavior of a Morse oscillator coupled to a bath of harmonic oscillators and driven by an infrared laser. The purpose is to compare quantum behavior of such a system with the classical and semiclassical solutions of a driven damped Morse oscillator. The general picture coming out of classical and semiclassical studies is that the total energy of the Morse oscillator in steady states or steady oscillations show jump and hysteretic behavior, as expected in a cusp catastrophe, when laser intensity or frequency is varied. Along with the bistable behavior we have also found period-doubling bifurcations leading to chaos on one or both branches. In the quantum treatment a generalized master equation is derived from the Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an SU(2) algebra. This algebra and the Markoffian approximation allows us to derive the equation of motion of the reduced density matrix, which is solved numerically. Bistable behavior does not exist in this quantum treatment, but solutions do show jump behavior and bifurcation phenomenon. Experimental implications of these studies are also discussed.
We have studied quantum-mechanical behavior of a driven Morse oscillator coupled to a bath of harmonic oscillators. The purpose is to compare the quantum behavior of such a system with the classical solutions of a driven damped Morse oscillator. We start with the quantum Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an su(2) Lie algebra. This algebra and the Markovian approximation allow us to derive the generalized master equation (for the reduced density matrix of the Morse oscillator), which contains level-dependent energy and phase relaxation terms. We have numerically integrated the differential equations of the matrix elements to obtain the time evolution of the reduced density matrix and the energy of the Morse oscillator. We show that the energy of the Morse oscillator in general varies with time and eventually reaches an asymptotic oscillatory state. The mean energy value of the asymptotic oscillatory state is studied as a function of the relaxation rates and laser frequency and amplitude. Vibrational distributions have also been found as functions of laser frequency and amplitude. The shape of the distribution changes gradually as laser amplitude increases; it first peaks at the ground state at a small field amplitude, then it peaks at some excited state at a large field amplitude, and for an amplitude in between it can have two maxima. This bimodal vibrational distribution reflects the bistability observed in the classical and semiclassical models. Thus bistability exists in quantum results, not as a hysteresis loop, but as a bimodal distribution. Finally, we show that as laser intensity increases, the time series of the oscillator energy evolves from regular to quasiperiodic behavior, and eventually a chaotic-looking series appears.
The comprehensive conditions concerning the harmonic limit of the Morse oscillator are investigated in this paper. These conditions extend the over-simple “classical” condition, which results by truncating the expansion of the Morse potential function in the power series of the coordinate close to the equilibrium position. The validity of the obtained results is tested by calculating the harmonic limit of the expected values of some characteristic physical observables of the Morse oscillator systems.