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QUANTUM OSCILLATOR UNDER EXTERNAL IMPULSE FORCE

A. Sanin, Andrey T. Bagmanov

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Abstract. In this contribution dynamics of quantum oscillator under external impulse force is investigated. The oscillator with symmetric polynomial potential in the well of impenetrable walls is considered. The potential consists of quadratic and cubic terms; the cubic term de-pends on modulus of coordinate. Influence of walls and anharmonicity on vibrations is dis-cussed. Properties and peculiarities of resonances have been studied in context of non-stationary Schrödinger’s equation at specified initial conditions. Solutions for the probability density, expectation coordinate as function of time; Fourier’s spectra were analyzed in detail.

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Abstract. In this contribution dynamics of quantum oscillator under external impulse force is investigated. The oscillator with symmetric polynomial potential in the well of impenetrable walls is considered. The potential consists of quadratic and cubic terms; the cubic term de-pends on modulus of coordinate. Influence of walls and anharmonicity on vibrations is dis-cussed. Properties and peculiarities of resonances have been studied in context of non-stationary Schrödinger’s equation at specified initial conditions. Solutions for the probability density, expectation coordinate as function of time; Fourier’s spectra were analyzed in detail.
Quantum dynamics of wave packet in one-dimensional system with distributed potential and dissipation is investigated in context of Schrodinger-Langevin-Kostin equation. Quadratic potential and double well potential bounded by external impenetrable walls are represented as examples. Numerical simulations are carried out for initial Gaussian packet at different values of dissipation parameters. For the quantum oscillator with dissipation the asymptotic dynamical limit in time is established. The mean values of coordinate and velocity tend to zero but uncertainty relation tends to minimal value 0.5 (in atomic units). Oscillations and resonances at external time-dependent action were studied. Dynamical properties of system with double well potential are investigated at different frequencies of external action. Fourier spectra of mean coordinate are characterized by sharply expressed singular components.
The quantum dynamics of a hydrogen-like atom confined in one-dimensional box with oscillating walls is studied.The description of the system is reduced to a one-dimensional Schrödinger equation for Coulomb potential with time-dependent boundary conditions, which is solved numerically.Using the obtained solution, the average kinetic energy and binding energies are calculated as a function of time.It is found that both the average kinetic energy and the binding energies are periodic in time with the period depending on the wall's oscillation parameters.The probability density is also analyzed as a function of time and coordinate.
Dynamics of wave packets in quantum systems subjected to a friction force and impulse feedback is investigated in context of Schrodinger-Langevin-Kostin (SchLK) equation. The quantum systems with a quadratic potential bounded by impenetrable walls of a well are presented as examples. Numerical simulations are carried out for initial Gaussian packet at varied values of a friction coefficient and amplitudes of the impulse feedback. Informative and structural properties of the wave packets are analyzed by using Fourier transform of dynamical variables and their means, of autocorrelation Nauenberg's function. Coherent oscillations of the wave packets and transition to chaotic dynamics are found. Influence of friction on oscillatory dynamics is discussed.
Quantum wave dynamics in systems with potential wells, bathers and outer impenetrable walls has been studied using solutions of time-dependent Schrodinger equation. Numerical integration of Schrodinger equation, calculation of dynamical variables and its average values, Fourier-spectra were carried out for different initial conditions and different relations between wave packet energy and bather height. In non-symmetric wells the oscillations of wave packets are accompanied by multistep discrete tunneling of probability fluid from well through barrier. In double well potential (symmetric wells) the temporal scales of oscillations and tunneling were investigated in details. Influence of additional time-dependent potential on quantum wave packet dynamics in above listed systems is also discussed.
Dynamical tunneling of a quantum wave packet in the system with distributed potential and impenetrable walls is discussed. This potential is specified as algebraic difference of quadratic and cubed functions. It is limited on finite length and incorporates local minimum and maximum. Potential distribution in the vicinity of local minimum forms asymmetric well, and barrier originates into domain with maximum. Initial Gaussian wave packet is located into the potential minimum and tunneling comes through potential maximum. Numerical integration of Schrodinger equation was carried out at zero conditions on system walls. For high barrier, the wave packet oscillates in the well for a long time and its small portions passes through the barrier. If the barrier height is small, tunneling takes place in a short time interval. Expectation coordinate and field velocity, probability density at separate points as function of time, control of uncertainty relation and normalization condition were calculated in detail.