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Ibn AL- Haitham Journal For Pure and Applied Science

Numerical Calculations and Time Evolution of Coherent States Wave Functions of Charged Oscillator in Magnetic Field

W.R. Mohsen

Published Jun 1, 2019• 0 citations
OpenAlex

Abstract

The wave functions of the coherent states of the charged oscillator in magnetic field are obtained via a canonical transformation. The numerical calculations of these functions are made and then the space and time plots are obtained. It was shown that these states are Gaussians distributions of widths vary periodically in an opposite way with their peaks. We interpret that is due to the mutual actions of the spreading effect of the wave packet and the reaction of the magnetic field.

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The wave functions of the coherent states of the charged oscillator in magnetic field are obtained via a canonical transformation. The numerical calculations of these functions are made and then the space and time plots are obtained. It was shown that these states are Gaussians distributions of widths vary periodically in an opposite way with their peaks. We interpret that is due to the mutual actions of the spreading effect of the wave packet and the reaction of the magnetic field.
Abstract We have calculated the effect of a magnetic field on the evolution of angular momentum eigenfunctions of a charged particle. An additional harmonic potential is supplemented to trap the wave packet. We find the probability density of the wave function is oscillating in the radial direction with a time period determined by the strength of the effective harmonic potential. When the magnetic field is along the z direction, if the initial wave function is an eigenfunction of L ˆ z , the probability density of the particle remains axis-symmetric. While for the case of an eigenfunction of L ˆ x , it is anisotropic in the x−y plane and rotates with a time period inverse proportional to the strength of the external magnetic field. We also extend the results in a phenomenological way to the case with an external magnetic field that varies harmonically in time.
The dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a time-dependent external magnetic field is investigated. We used canonical transformation approach for the classical treatment of the system, whereas unitary transformation approach is used when managing the system in the framework of quantum mechanics. For both approaches, the original system is transformed to a much more simple system that is the sum of two independent harmonic oscillators which have time-dependent frequencies. We therefore easily identified the wave functions in the transformed system with the help of invariant operator of the system. The full wave functions in the original system is derived from the inverse unitary transformation of the wave functions associated to the transformed system.
We solve the quantum mechanical problem of the motion of charged particle in a quadrupole field that varies arbitrarily with time. The time-dependent wave functions are shown to be in a simple one-to-one correspondence with the wave functions of an ordinary static-field harmonic oscillator. We define and generate a set of coherent states for the charged particle that provide Gaussian localization about the classical particle trajectories. When the field variation is periodic in time, these minimum uncertainty states are shown to pulsate periodically in size and in phase. Under the action of external fields, coherent states retain their coherence. We discuss the way in which the Kapitza pseudopotential approach approximates the present exact results.