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Chinese Journal of Physics

Gaussian Wave Packet for a Time-Dependent Harmonic Oscillator Model of a Charged Particle in a Variable Magnetic Field

Salah Menouar, Mustapha Maamache, Jeong Ryeol Choi

Published Aug 1, 2011• 5 citations
OpenAlex

Abstract

We obtained the exact solution of the Schrodinger equation for a time-dependent harmonic oscillator model of a charged particle in a variable magnetic field, which is the semiclassical Gaussian wave packet centered around the classical guiding trajectory (p(subscript i), q(svbscript i)). The Gaussian wave packet presented here is well-localized in space and useful for determining various features of the classical-like wave packet.

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We obtained the exact solution of the Schrodinger equation for a time-dependent harmonic oscillator model of a charged particle in a variable magnetic field, which is the semiclassical Gaussian wave packet centered around the classical guiding trajectory (p(subscript i), q(svbscript i)). The Gaussian wave packet presented here is well-localized in space and useful for determining various features of the classical-like wave packet.
In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of a electro-magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac--Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth order Runge--Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.
In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of charged particles under the influence of an electro-magnetic field. The solution of the time-dependent Schrödinger equation is approximated by a single Gaussian wave packet via the time-dependent Dirac–Frenkel variational principle. For the approximation we use ordinary differential equations of motion for the parameters of the variational solution and extend the second-order Boris algorithm for classical mechanics to the quantum mechanical case. In addition, we propose a modified version of the classical fourth-order Runge–Kutta method. Numerical experiments explore parameter convergence and geometric properties. Moreover, we benchmark against the analytical solution of the Penning trap.
In the present paper we consider the semiclassical magnetic Schrödinger equation, which describes the dynamics of particles under the influence of a magnetic field. The solution of the Schrödinger equation is approximated by Gaussian wave packets via the time-dependent variational formulation by Dirac and Frenkel. For the numerical approximation we derive ordinary differential equations for the parameters of the variational solution. Moreover, we prove L2L^2L2-error bounds and observable error bounds for the approximating Gaussian wave packet.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.