Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Quantum Dynamics of Anharmonic Oscillators. II. Systems Having One and Two Degrees of Freedom

Roger C. Baetzold, Charlotte T. Tahk, David J. Wilson

Published Dec 1, 1966• 15 citations
Publisher

Abstract

Recent calculations on two simple anharmonic oscillators have indicated significant discrepancies between the evolution with time of the quantum and classical expectation values of the coordinate. These results cast some doubt on the propriety of classical-mechanical theories of unimolecular rates. The present paper extends the quantum calculations to two simple systems having two degrees of freedom and to the semi-infinite square-well oscillator.

DOI
10.1063/1.1727480

Related papers

Quantum Dynamics of Anharmonic Oscillators. I. Simple Examples

Elliott B. Alterman, Charlotte T. Tahk, David J. Wilson

The dynamical behavior of two anharmonic oscillators, having a linear and a square-well potential, respectively, is studied. The expectation values of the coordinate and the mean spread of the oscillating wave packets are calculated as functions of time and presented graphically. Significant departures from the behavior of the corresponding classical systems are observed. These results cast some doubt on the use of classical-mechanical models in the theory of unimolecular rates.

The Journal of Chemical Physics•Jan 15, 1966•14 citations

Quantum Dynamics of Anharmonic Oscillators. III. The Morse Oscillator

Paul F. Endres, David J. Wilson

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 Journal of Chemical Physics•Jan 15, 1967•26 citations

An analysis of states in the phase space: the anharmonic oscillator

Sebastiano Tosto

The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a complex system of energy levels dependent upon the number of anharmonic terms that characterize the oscillator.

arXiv:1105.3615•May 18, 2011•2 citations

Average value of position for the anharmonic oscillator: Classical versus quantum results

Richard W. Robinett

The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.

American Journal of Physics•Mar 1, 1997•15 citations

On the energy levels of an isotropic anharmonic oscillator

M. Lakshmanan, P. Kaliappan

An approximate implicit energy level expression is obtained for a spherically symmetric anharmonic oscillator with three degrees of freedom, by using a semi-classical, WKB type Bohr-Sommerfeld quantisation rule.

Journal of Physics A Mathematical and General•Sep 1, 1980•12 citations

Coupled Anharmonic Oscillators. II. Unequal-Mass Case

T. Banks, Carl M. Bender

We develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1973•137 citations
Recent calculations on two simple anharmonic oscillators have indicated significant discrepancies between the evolution with time of the quantum and classical expectation values of the coordinate. These results cast some doubt on the propriety of classical-mechanical theories of unimolecular rates. The present paper extends the quantum calculations to two simple systems having two degrees of freedom and to the semi-infinite square-well oscillator.
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 paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a complex system of energy levels dependent upon the number of anharmonic terms that characterize the oscillator.
The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.
We develop a general formalism for calculating the large-order behavior of perturbation theory for quantized systems of unequal-mass coupled anharmonic oscillators. Our technique is based on a generalization of the semiclassical approximation which was used to study equal-mass oscillators in the first paper of this series. The unequal-mass problem is much more difficult because the path which minimizes the classical action is not a straight line. Assuming that this tunneling path is known, we derive a general expression for the physical-optics approximation to the wave function of a tunneling particle. This derivation rests on the construction of a WKB approximation in curved space. We thus completely reduce the general quantum problem to the much simpler classical one of determining the path. Then we present a perturbation scheme for finding the classical path for systems of oscillators whose masses only differ by a small amount. Finally, we illustrate our techniques by solving a two-mode unequal-mass oscillator and comparing these results with a computer calculation. Our theoretical predictions and numerical calculations agree.