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Physica Scripta

Fisher information of quantum damped harmonic oscillators

V. Aguiar, I. Guedes

Published Mar 27, 2015• 23 citations

Abstract

We calculate the time-dependent Fisher information in position and momentum for the lowest lying state of two classes of quantum damped (Lane–Emden (LE) and Caldirola–Kanai (CK)) harmonic oscillators. The expressions of and are written in terms of a c-number quantity satisfying a nonlinear differential equation. Analytical solutions of were obtained. For the LE and CK oscillators, we observe that increases while decreases with increasing time. The product increases and tends to a constant value in the limit for the LE oscillator, while it is time-independent for the CK oscillator. Moreover, for the CK oscillator the product decreases as the damping increases. Relations among the Fisher information, Leipnik and Shannon entropies, and the Stam and Cramer–Rao inequalities are given. A discussion on the squeezing phenomenon in position for the oscillators is presented.

DOI: 10.1088/0031-8949/90/4/045207

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We calculate the time-dependent Fisher information in position and momentum for the lowest lying state of two classes of quantum damped (Lane–Emden (LE) and Caldirola–Kanai (CK)) harmonic oscillators. The expressions of and are written in terms of a c-number quantity satisfying a nonlinear differential equation. Analytical solutions of were obtained. For the LE and CK oscillators, we observe that increases while decreases with increasing time. The product increases and tends to a constant value in the limit for the LE oscillator, while it is time-independent for the CK oscillator. Moreover, for the CK oscillator the product decreases as the damping increases. Relations among the Fisher information, Leipnik and Shannon entropies, and the Stam and Cramer–Rao inequalities are given. A discussion on the squeezing phenomenon in position for the oscillators is presented.

We obtain the exact wave functions of two Lane-Emden-type Kanai-Caldirola (LE-KC) oscillators by using the dynamical invariant method. To do so, we analytically solve the respective Milne-Pinney equation for each oscillator. We also calculate the uncertainty product, the transition probability and the quantum-mechanical energy expectation value for each oscillator. We compare the results with those of the well-known KC oscillator. The quantum-mechanical energy expectation value of the KC oscillator goes to zero faster than that of the LE-KC oscillators, indicating that the KC system is more damped than the LE-KC ones.

Abstract We first present the analytical solutions to the Schrödinger equation with the parity-restricted harmonic oscillator V ( x ) = 1 2 m ω 2 x 2 ( x > 0 ) and then calculate the Shannon information entropies S x n and S p n both in position space and in momentum space. It is interesting to find that the variation of the Shannon information entropy S p n in momentum space is different from our previous studies, i.e. the S p n first increases with the quantum number n and then decreases with the number n . This is a new and abnormal phenomenon and may be explained by the parity-restricted system. The BBM inequality is verified to be saturated, but the sum of the entropies first increases with the number n and then decreases with it. The entropy densities ρ s ( x ) and ρ s ( p ) are also demonstrated. We find that the Fisher entropy is exactly given by I F = 8 n + 6 , n = 0 , 1 , 2 , … .

In this paper, we studied, at first, the influence of the energy-dependent potentials on the one-dimensionless Klein–Gordon oscillator. Then, the Shannon entropy and Fisher information of this system are investigated. The position and momentum information entropies for the low-lying states n = 0, 1, 2 are calculated. Some interesting features of both Fisher and Shannon densities, as well as the probability densities, are demonstrated. Finally, the Stam, Cramer–Rao and Bialynicki–Birula–Mycielski (BBM) inequalities have been checked, and their comparison with the regarding results have been reported. We showed that the BBM inequality is still valid in the form [Formula: see text], as well as in ordinary quantum mechanics.

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrodinger equation for the Caldirola-Kanai time-dependent forced harmonic oscillator in terms of the solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the quantum fluctuations in coordinate and momentum as well as the uncertainty product. It is also shown that the quantum fluctuations and the width of the Gaussian packet do not depend on the external force.