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

Oleksandr Serhiyovych Mazmanishvili

Published Oct 25, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.46813/2024-153-166

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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.
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.
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.
Localization of relativistic particles have been of great research interests over many decades. We investigate the time evolution of the Gaussian wave packets governed by the one dimensional Dirac equation. For the free Dirac equation, we obtain the evolution profiles analytically in many approximation regimes, and numerical simulations consistent with other numerical schemes. Interesting behaviors such as Zitterbewegung and Klein paradox are exhibited. In particular, the dispersion rate as a function of mass is calculated, and it yields an interesting result that super-massive and massless particles both exhibit no dispersion in free space. For the Dirac equation with random potential or mass, we employ the Chebyshev polynomials expansion of the propagator operator to numerically investigate the probability profiles of the displacement distribution when the potential or mass is uniformly distributed. We observe that the widths of the Gaussian wave packets decrease approximately with the power law of order o(s−ν)o(s^{-ν})o(s−ν) with $\frac{1}{2}
The quantum wave-function of a massive particle with small initial uncertainties (consistent with the uncertainty relation) is believed to spread very slowly, so that the dynamics is deterministic. This assumes that the classical motions for given initial data are unique. In fluid turbulence non-uniqueness due to "roughness" of the advecting velocity field is known to lead to stochastic motion of classical particles. Vanishingly small random perturbations are magnified by Richardson diffusion in a "nearly rough" velocity field so that motion remains stochastic as the noise disappears, or classical spontaneous stochasticity, . Analogies between stochastic particle motion in turbulence and quantum evolution suggest that there should be quantum spontaneous stochasticity (QSS). We show this for 1D models of a particle in a repulsive potential that is "nearly rough" with V(x)∼C∣x∣1+αV(x) \sim C|x|^{1+α}V(x)∼C∣x∣1+α at distances ∣x∣≫ℓ|x|\gg \ell∣x∣≫ℓ , for some UV cut-off ℓ\ellℓ, and for initial Gaussian wave-packet centered at 0. We consider the WKB limit with ℏ/m→0\hbar/m \to 0ℏ/m→0, then position-spread σ→0σ\to 0σ→0. The limit of the position density is non-deterministic, with equal probabilities of the particle following either of two, non-unique classical solutions and with "Richardson-like" spreading of the wave-packet. The splitting of the wave-packet occurs in a very short time t∼(ℓ1+α/C)1/2t\sim(\ell^{1+α}/C)^{1/2}t∼(ℓ1+α/C)1/2 for small ℓ\ellℓ. QSS also occurs in the scattering of a wave-packet off the rough potential, for careful fine-tuning. We also consider other semi-classical limits by a numerical solution of the Schrödinger equation, observing QSS both in position- and momentum-space. Although the wave-function remains split into two widely separated branches in the classical limit, rapid phase oscillations within each branch prevent any coherent superposition. These effects should be observable in laboratory experiments.
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 discuss the time evolution of the wave function which is the solution of a stochastic Schrödinger equation describing the dynamics of a free quantum particle subject to spontaneous localizations in space. We prove global existence and uniqueness of solutions. We observe that there exist three time regimes: the collapse regime, the classical regime and the diffusive regime. Concerning the latter, we assert that the general solution converges almost surely to a diffusing Gaussian wave function having a finite spread both in position as well as in momentum. This paper corrects and completes earlier works on this issue.