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arXiv:cond-mat/0510812

Exact approaches to charged particle motion in a time-dependent flux-driven ring

Pi‐Gang Luan, Chi‐Shung Tang

Published Oct 30, 2005• 0 citations
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Abstract

We consider a charged particle which is driven by a time-dependent flux threading a circular ring system. Various approaches including classical treatment, Fourier expansion method, time-evolution method, and Lewis-Riesenfeld method are used and compared to solve the time-dependent problem. By properly managing the boundary condition of the system, a time-dependent wave function of the charged particle can be obtained by using a non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. The eigenfunction of the linear invariant can be realized as a Gaussian-type wave packet with a peak moving along the classical angular trajectory, while the distribution of the wave packet is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. In this topologically nontrivial system, we find that although the classical trajectory and angular momentum can determine the motion of the wave packet; however, the peak position is no longer an expectation value of the angle operator. Therefore, in such a system, the Ehrenfest theorem is not directly applicable.

DOI
10.48550/arxiv.cond-mat/0510812

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We consider a charged particle which is driven by a time-dependent flux threading a circular ring system. Various approaches including classical treatment, Fourier expansion method, time-evolution method, and Lewis-Riesenfeld method are used and compared to solve the time-dependent problem. By properly managing the boundary condition of the system, a time-dependent wave function of the charged particle can be obtained by using a non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. The eigenfunction of the linear invariant can be realized as a Gaussian-type wave packet with a peak moving along the classical angular trajectory, while the distribution of the wave packet is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. In this topologically nontrivial system, we find that although the classical trajectory and angular momentum can determine the motion of the wave packet; however, the peak position is no longer an expectation value of the angle operator. Therefore, in such a system, the Ehrenfest theorem is not directly applicable.
We consider a charged particle driven by a time-dependent flux threading a quantum ring. The dynamics of the charged particle is investigated using a classical treatment, a Fourier expansion technique, a time-evolution method, and the Lewis-Riesenfeld approach. We have shown that, by properly managing the boundary conditions, a time-dependent wavefunction can be obtained using a general non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. It is shown that the linear invariant eigenfunction can be realized as a Gaussian-type wavepacket with a peak moving along the classical angular trajectory, while the distribution of the wavepacket is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. From the topologically nontrivial nature as well as the classical trajectory and angular momentum, one can determine the dynamical motion of the wavepacket. It should be noted that the peak position is no longer an expectation value of the angle operator, and hence the Ehrenfest theorem is not directly applicable in such a topologically nontrivial system.
We reexamine the general solution of a Schr\"odinger equation in the presence of a time-dependent linear potential in configuration space based on the Lewis-Riesenfeld framework. For comparison, we also solve the problem in momentum space and then Fourier transform the solution to get the general wave function. Appropriately choosing the weight function in the latter method, we can obtain the same wave function as the former method. It is found that a non-Hermitian time-dependent linear invariant can be used to obtain Gaussian-type wave-packet solutions of the time-dependent system. This operator is a specific linear combination of the initial momentum and initial position operators. This fact indicates that the constants of integration such as the initial position and initial momentum that determine the classical motion play important roles in the time-dependent quantum system. The eigenfunction of the linear invariant is interpreted as a wave packet with a ``center of mass'' moving along the classical trajectory, while the ratio between the coefficients of the initial position and initial momentum determines the width of the wave packet.
We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.
Operator-difference multilayer schemes for solving the time-dependent Schrödinger equation up to sixth order of accuracy in the time step are presented. Reduced schemes for solving a set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised using expansion of a wave packet over the set of appropriate basis angular functions. Further discretization of the resulting problem is realized by means of the finite-element method. The convergence of the expansion with respect to the number of basis functions and the efficiency of the numerical schemes are demonstrated in the exactly solvable model of an electric-field-driven two-dimensional oscillator (or a charged particle in a constant uniform magnetic field), in which we explicitly observed an effect of the periodical focusing and defocusing of the probability density flux.