Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

RESONANT SPREAD WAVE FUNCTION IN PARABOLIC POTENTIAL

Александр Сергеевич Мазманишвили

Published Aug 15, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.46813/2019-122-100

Related papers

STOCHASTIC DISTURBANCE BY “WHITE” NOISE WAVE FUNCTIONS IN THE PARABOLIC POTENTIAL AND STATISTICAL WAVE PACKET LIFETIME DISTRIBUTION

Oleksandr Serhiyovych Mazmanishvili

The work examines the parabolic potential, which as the whole is subject to the stochastic influence of the random process of “white” noise, and the dynamics of the wave function of a particle in it. Based on the found solu- tions to the nonstationary Schrödinger equation, the time evolution of the wave function is considered. The problem of destroying the wave packet of a particle is formulated and analyzed, which is realized when the condition of achieving the dispersion of the packet of a given size is met. For the random variable – time to destruction – the probability distribution density is given.

•Oct 25, 2024•0 citations

Dynamics of Quantum Particles in Perturbed Parabolic 2d Potential

NSC ", Александр Сергеевич Мазманишвили, , 1, Academicheskaya st., 61108, Kharkov, Ukraine +1 more

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.

Journal of Nano- and Electronic Physics•Jan 1, 2016•2 citations

INFLUENCE OF THE WHITE NOISE ON QUANTUM DYNAMICS OF THE WAVE FUNCTION IN THE TIME-PERIODIC HARMONIC POTENTIAL

Oleksandr Serhiyovych Mazmanishvili, Vladimir I. Dubinko, Klee Irwin

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.

Voprosy atomnoj nauki i tehniki. Seriâ âderno-fizičeskie issledovaniâ•Jun 12, 2026•0 citations

Nonspreading wave packets in a general potential V(x,t) in one dimension

Chyi-Lung Lin, Meng-Jie Huang, Te-Chih Hsiung

We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential $\textmd{V}(x,t)$. The essential ingredients of a nonspreading wave packet, the shape function $f(x)$, the motion $d(t)$, the phase function $ϕ(x,t)$ are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function $f(x)$ is the eigenfunction of the time independent Schrödinger equation with an effective potential $V_{\textmd{eff}}$ and an energy $E_{\textmd{eff}}$. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.

arXiv:0809.4105•Sep 24, 2008•1 citation

Linear Potentials, Airy Wave Packets and Airy Transform

G. Dattoli, Жуковский Константин Владимирович

The solution of the Schrodinger equation with a linear potential is considered. We use algebraic methods to obtain the explicit form of the solution for the explicitly time dependent Hamiltonian and discuss the general conditions which allow us to get solutions in terms of the Airy functions, yelding non spreading wave packets. We analyze the relevant physical meaning of these solutions and give examples of their applications. We discuss the analogy between Airy and Gauss-Weirestrass transform.

arXiv:1010.1678•Oct 8, 2010•2 citations

Free Particle, Wave Packet and Dynamics, Quantum Dots, Defects and Traps

Duncan G. Steel

Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.

•Apr 30, 2021•0 citations
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.
The work examines the parabolic potential, which as the whole is subject to the stochastic influence of the random process of “white” noise, and the dynamics of the wave function of a particle in it. Based on the found solu- tions to the nonstationary Schrödinger equation, the time evolution of the wave function is considered. The problem of destroying the wave packet of a particle is formulated and analyzed, which is realized when the condition of achieving the dispersion of the packet of a given size is met. For the random variable – time to destruction – the probability distribution density is given.
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.
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.
We discuss nonspreading wave packets in one dimensional Schrödinger equation. We derive general rules for constructing nonspreading wave packets from a general potential V(x,t)\textmd{V}(x,t)V(x,t). The essential ingredients of a nonspreading wave packet, the shape function f(x)f(x)f(x), the motion d(t)d(t)d(t), the phase function ϕ(x,t)ϕ(x,t)ϕ(x,t) are derived. Since the form of the shape of a nonspreading wave packet does not change in time, the shape equation should be time independent. We show that the shape function f(x)f(x)f(x) is the eigenfunction of the time independent Schrödinger equation with an effective potential VeffV_{\textmd{eff}}Veff​ and an energy EeffE_{\textmd{eff}}Eeff​. We derive nonspreading wave packets found by Schrödinger, Senitzky, and Berry and Balazs as examples. We show that most stationary potentials can only support stationary nonspreading wave packets. We show how to construct moving nonspreading wave packets from time dependent potentials, which drive nonspreading wave packets into an arbitrary motion.
Abstract This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum well. A free particle can exist over all space or can be localized in a wave package. The wave packet is a coherent superposition of the plane waves that make up the wave function that localizes the particle because of constructive and destructive interference. The wave packet spreads out in time because the waves leading to constructive interference get out of phase. In Chapter 2, the particle was localized by a quadratic potential energy. Here, the potentials are described as piecewise constant. The approach is based on assuming a one-dimensional space, x, which is relevant to many problems in the laboratory. The solution is easily generalized to higher dimensions (x-y or x-y-z), but the physics remains the same. The objective is to understand the shape of the eigenfunctions in space and to be able to relate this to the probability density of locating the particle, as well as understanding the relevance of these systems to today’s technology.