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Journal of Nano- and Electronic Physics

Dynamics of Quantum Particles in Perturbed Parabolic 2d Potential

NSC ", Александр Сергеевич Мазманишвили, , 1, Academicheskaya st., 61108, Kharkov, Ukraine, I. A. Knyaz’

Published Jan 1, 2016• 2 citations
Publisher

Abstract

2d quantum-mechanical problem of the time evolution of a particle in a quadratic potential is studied. We suppose that the center of the potential is displaced in arbitrary way in time. An analytical expression for the wave function in arbitrary instant time was built. It is shown the dynamic shift of the center of the potential doesn’t change the variance. Moreover, the system can exhibit the resonance: when the frequency of the potential perturbation approaches to the natural frequency the amplitude of the wave packet of particle is increased.

DOI
10.21272/jnep.8(4(1)).04014

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2d quantum-mechanical problem of the time evolution of a particle in a quadratic potential is studied. We suppose that the center of the potential is displaced in arbitrary way in time. An analytical expression for the wave function in arbitrary instant time was built. It is shown the dynamic shift of the center of the potential doesn’t change the variance. Moreover, the system can exhibit the resonance: when the frequency of the potential perturbation approaches to the natural frequency the amplitude of the wave packet of particle is increased.
In this paper, we consider the parabolic potential, which as a whole is subject to dipole or quadrupole action (parametric resonance), which periodically changes with time, and the dynamics of the wave function of a particle. Based on the solutions found for the nonstationary Schrödinger equation, algorithms for calculating the dynamics of the wave function are constructed. The evolution of the wave function of a particle is analyzed. Asymptotic solutions of the equation of motion are given, using which the main characteristics of the wave packet are obtained. For selected types of potential perturbations, examples of the evolution of the wave function are given.
We investigate the quantum dynamics of a particle in a one-dimensional parabolic potential subject to both timeperiodic driving and stochastic perturbations modeled as white noise. Using the non-stationary Schrödinger equation and the Green’s function formalism, we derive analytical expressions for the time evolution of the wave function and its statistical moments. It is shown that, due to the quadratic nature of the potential, the wave function retains a Gaussian form, allowing a reduced description in terms of its first and second moments. Under resonant periodic driving, the mean energy exhibits quadratic growth in time. In the presence of white noise, the wave packet undergoes diffusive spreading and loss of coherence. For combined deterministic and stochastic driving, we identify a characteristic decoherence (delocalization) time, which scales inversely with the noise intensity. The results are relevant to quantum transport, driven systems, and tunneling processes in fluctuating environments.
The behavior of a particle moving according to the laws of quantum mechanics in a field, which potential changes over time, is studied. The method of unitary transformations for solving the temporal Schrödinger equation is considered. The reduced Hamiltonian of the system of a quantum­-mechanical particle in time­-dependent potential is obtained, as well as the total operator of the evolution of such a system. We find a new version of the unitary transformation, which, compared to the known ones, simplifies solving the problem in analytical form. This transformation associates the linear potential quantum model with time-­dependent parameters and free­particle system. Based on the constructed evolution operator, we consider the wave packet dispersion. It is shown that the wave packet dispersion for this model is the same as that for a free particle. This conclusion is the general result of the behavior of a quantum­mechanical particle moving in the field of a time­dependent linear potential.
This work is a simple example of the quantum dynamics of a particle in a disordered system in one dimension.In particular, we illustrate numerically the phenomenon of Anderson localisation of a wave packet using a basic model constructed with small random rectangular potential barriers.Also, we study the dynamics of a quantum particle in a disordered potential formed by an harmonic oscillator perturbed by random rectangular barriers.To show the effects of disorder on the dynamics of the system, we compare the time evolution of the wave function of the unperturbed harmonic oscillator with the wave function of the disordered system.We do this by taking the scalar product between the unperturbed and perturbed wave functions at each timestep for different values of the perturbation parameters affecting the disordered wave packet.
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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.