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College Physics

The classical characteristics of movement of one-dimensional quantum forced oscillator

Xia Xiao-jian

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The one-dimensional classical forced oscillator is quantized.The exact solution of time evolution operator of the system is obtained.The quasi-probability distribution function of phase space is used to investigate the quantum characteristics of one-dimensional forced oscillator.The results show that the Wigner function is two-dimensional Gaussian wave packet from the initial vacuum state.The central coordinate and momentum of the Gaussian wave packet move in the same manner as one-dimensional classical harmonic oscillator.The amplitude and phase of the centre of the wave packet are modulated by the force.

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The one-dimensional classical forced oscillator is quantized.The exact solution of time evolution operator of the system is obtained.The quasi-probability distribution function of phase space is used to investigate the quantum characteristics of one-dimensional forced oscillator.The results show that the Wigner function is two-dimensional Gaussian wave packet from the initial vacuum state.The central coordinate and momentum of the Gaussian wave packet move in the same manner as one-dimensional classical harmonic oscillator.The amplitude and phase of the centre of the wave packet are modulated by the force.
With the help of principle of superposition in quantum mechanics and the operator splitting method,we study the movement of the center position of the initial Gaussian wave packet in one-dimensional harmonic oscillator potential.The results show that its quantum evolution performs characteristics of a classical harmonic oscillator;the initial location and the initial momentum of the wave packet play a similar role as the classical oscillator.Our results should be helpful to understand the Gaussian wave packet method,which is commonly applied in the research on complex quantum system.
The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.
New exact and asymptotic results for a quantum inverted oscillator, driven by the variable external force, are presented. To illustrate the advantages of our approach, we applied the obtained propagator to the descriptions of evolution the initial Gaussian wave packet under arbitrary time-dependent force or a δ-pulse action. The alternative problem of tunneling a particle through a parabolic barrier under the influence of a low-frequency harmonic force we solve in the quasistatic approximation. Going to the Heisenberg picture in the Caldeira-Leggett model, we describe the influence of a time-dependent force on an open inverted oscillator. The resulting dynamics is presented in the form of combination the evolution of the average value of a coordinate operator and the stochastic diffusion, being induced the wave packet spreading and the Brownian motion.
Abstract Some results of computer simulation of the behavior of a one‐dimensional quantum mechanical oscillator are reported in this article. This harmonic oscillator comprises a particle trapped within a hyperbolic potential V (x) = x2. Further, a perturbation potential function V′(x, t) was superposed upon the hyperbolic potential in order to induce a quantum mechanical transition. This perturbation function V′(x, t) is a function of both of space and time variables, and is set to represent a wave packet that is enveloped by a Gaussian bell‐shaped curve. A wave that probably has an appropriate wave number and angular frequency was inputted into the expression for the wave packet. In the initial phase, while the harmonic oscillator was allowed to oscillate almost freely, the wave packet was allowed to approach the harmonic oscillator. In the middle phase, the wave packet passes through the harmonic oscillator, affecting the shape of the quantum mechanical wave that represents the physical state of the system. In the last phase, when the wave packet left the system of the harmonic oscillator, the system settled onto an energetically stable state. The main objective of the simulation was to simulate the instance of a quantum mechanical transition from one eigenstate to another. After several trials, it was found that the perturbation function consisting of a complex function was, at least superficially, able to cause one desired transition, that is, a transition from one eigenstate to another eigenstate. By using such a complex perturbation function, a transition from the first excited state to the ground state was observed to occur. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
Using the trial function method, exact wave functions of the time-dependent driven generalized harmonic oscillator (TDGHO) were derived. The wave functions are wave packets with the center moving according to the classical equation of motion. Correspondingly, the geometrical phase is divided into two parts with one part generated by the shape change of the wave packet and the other part by the motion of the wave packet center. The exact classical solution as well as the classical phase was obtained from a quantum description by the Heisenberg correspondence principle. Finally, the relationship between the quantum and classical phases was achieved.
Wigner phase space quasi-probability distribution function is a Fourier transform related to a given quantum mechanical wave function. It is shown that for the wave functions of type ψ(q)=e−aq2ϕ(q)ψ(q)=e^{-aq^2}ϕ(q)ψ(q)=e−aq2ϕ(q), the Wigner function can be defined in terms of differential operators acting on a given function, independently from the integral formula which appears in the standard definition. Gaussian wave packet, harmonic and singular oscillators are given as the examples.