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

On the bilinear approximation for time-dependent Hamiltonians

Julián Echave, Francisco Marcelo Fernández, Eduardo Alberto Castro

Published Jan 15, 1990• 4 citations
Publisher

Abstract

The transition probabilities for a one-dimensional harmonic oscillator with a time-dependent interaction potential are obtained. Two different bilinear approximations to the interaction potential are obtained by expanding it in Taylor series around two reference coordinate values, namely, the equilibrium position q0=0 and the expectation value of the coordinate q0=〈q〉, keeping only first- and second-order terms. The effect of the choice of q0 on the accuracy of the results is investigated for several values of the interaction potential parameters. The model posed by the semiclassical approach to the collision between a structureless particle and a harmonic oscillator via an exponential interaction potential is chosen as an illustrative example. The expansion around 〈q〉 is found to be related to an iterative procedure, based on Lie algebraic methods, recently proposed to treat time-dependent quantum-mechanical problems.

DOI
10.1063/1.458127

Related papers

On the Exponential Form of Time-Displacement Operators in Quantum Mechanics

Philip Pechukas, John C. Light

We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.

The Journal of Chemical Physics•May 15, 1966•549 citations

Exact formulated approach to transition probabilities of linearly driven parametric oscillators

Wensen Liu

The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.

Physica Scripta•Mar 1, 2009•0 citations

Time-dependent perturbation and exact results for the periodically driven quantum harmonic oscillator

Russell Akridge

Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.

American Journal of Physics•Feb 1, 1995•5 citations

The quantum-mechanical harmonic oscillator

Francisco Marcelo Fernández, Eduardo Alberto Castro

The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.

CRC Press eBooks•Jan 16, 2020•0 citations

Time-independent variational approach to inelastic collisions of a particle with a harmonic oscillator

Yasutami Takada

A variational approach is applied to collisions of a particle having a mass m with a harmonic oscillator having a mass M in one dimension. We have used trial functions which become exact in the limit of either m/M\ensuremath{\gg}1 or m/M\ensuremath{\ll}1. We have derived exact expressions in the limit m/M\ensuremath{\rightarrow}0 for inelastic as well as elastic collisions to leading order in the parameter m/M for any impenetrable interaction potential.

Physical Review A•Jul 1, 1988•2 citations

Exponential time-evolution operator for the time-dependent harmonic oscillator

Francisco Marcelo Fernández

The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.

Journal of Mathematical Physics•Dec 1, 1987•16 citations
The transition probabilities for a one-dimensional harmonic oscillator with a time-dependent interaction potential are obtained. Two different bilinear approximations to the interaction potential are obtained by expanding it in Taylor series around two reference coordinate values, namely, the equilibrium position q0=0 and the expectation value of the coordinate q0=〈q〉, keeping only first- and second-order terms. The effect of the choice of q0 on the accuracy of the results is investigated for several values of the interaction potential parameters. The model posed by the semiclassical approach to the collision between a structureless particle and a harmonic oscillator via an exponential interaction potential is chosen as an illustrative example. The expansion around 〈q〉 is found to be related to an iterative procedure, based on Lie algebraic methods, recently proposed to treat time-dependent quantum-mechanical problems.
We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.
The transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.
Transition amplitudes and probabilities for the harmonic oscillator with a forcing function proportional to cos(ωt) beginning at time zero are calculated to lowest nonvanishing order using time-dependent perturbation theory. The results are compared with the exact amplitudes and probabilities. When the exact amplitude is expanded in a Taylor series in powers of the coupling constant, the individual terms turn out to be the perturbation amplitudes, showing that the complete series of perturbation amplitudes converges to the exact amplitude.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.