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Journal of Physics A Mathematical and General

Multidimensional wave packet motion in quadratic potentials with and without friction: oscillation and barrier penetration

Rainer W. Hasse

Published Jul 1, 1978• 24 citations
Publisher

Abstract

The author studies the motion of N-dimensional wave packets in quadratic potentials without friction and also under the influence of two different quantum frictional potentials. Differential equations for the coordinate and momentum uncertainties and for the correlations are presented and solutions for oscillatory motion and barrier penetration are given where possible. The barrier penetration probabilities are compared with each other.

DOI
10.1088/0305-4470/11/7/013

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The author studies the motion of N-dimensional wave packets in quadratic potentials without friction and also under the influence of two different quantum frictional potentials. Differential equations for the coordinate and momentum uncertainties and for the correlations are presented and solutions for oscillatory motion and barrier penetration are given where possible. The barrier penetration probabilities are compared with each other.
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.
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Using the phenomenological quantum friction models introduced by P. Caldirola [Nuovo Cimento 18, 393 (1941)] and E. Kanai [Prog. Theor. Phys. 3, 440 (1948)], M. D. Kostin [J. Chem. Phys. 57, 3589 (1972)], and K. Albrecht [Phys. Lett. B 56, 127 (1975)], we study quantum tunneling of a one-dimensional potential in the presence of energy dissipation. To this end, we calculate the tunneling probability using a time-dependent wave-packet method. The friction reduces the tunneling probability. We show that the three models provide similar penetrabilities to each other, among which the Caldirola-Kanai model requires the least numerical effort. We also discuss the effect of energy dissipation on quantum tunneling in terms of barrier distributions.
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.
An interesting analytical expression is obtained for the coordinate probability distribution of a wave packet transmitted through a potential barrier. In the large-time asymptotics the distribution is given in terms of the transmission amplitude and the phase-space quasidistribution of the initial (not necessarily pure) state. The distortion of a (broad) wave packet due to the barrier penetration is determined by two timelike parameters, given by the energy derivative of the complex transmission amplitude. The resulting change of the packet shape reveals a dependence on the detector position and incorporates a possible speed-up effect and a negative dispersion.