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Journal of Mathematical Physics

Entropy and information of a spinless charged particle in time-varying magnetic fields

V. Aguiar, I. Guedes

Published Sep 1, 2016• 30 citations
Publisher

Abstract

We calculate the Fisher information (Fr and Fp) and the Shannon entropies (Sr and Sp) of a spinless charged particle in three different configurations of time-varying magnetic field, B(t). To do so, we first obtain the exact wave functions for a general time-dependent system by using a dynamical invariant method. By considering the solutions for n = m = 0, we were able to obtain the expressions of Fr, Fp, Sr, and Sp in terms of a c-number quantity, ρ, which has to be a real solution of the Milne–Pinney equation. We observe that the inequality FrFp ≤ 16 holds for the systems considered. We also observed squeezing phenomenon in momentum or/and coordinate spaces with increasing time.

DOI
10.1063/1.4962923

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We calculate the Fisher information (Fr and Fp) and the Shannon entropies (Sr and Sp) of a spinless charged particle in three different configurations of time-varying magnetic field, B(t). To do so, we first obtain the exact wave functions for a general time-dependent system by using a dynamical invariant method. By considering the solutions for n = m = 0, we were able to obtain the expressions of Fr, Fp, Sr, and Sp in terms of a c-number quantity, ρ, which has to be a real solution of the Milne–Pinney equation. We observe that the inequality FrFp ≤ 16 holds for the systems considered. We also observed squeezing phenomenon in momentum or/and coordinate spaces with increasing time.
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In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.
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Abstract The Shannon entropy (S) and the Fisher Information (I) entropies are investigated for a generalized hyperbolic potential in position and momentum spaces. First, the Schrodinger equation is solved exactly using the Nikiforov‐Uvarov‐Functional Analysis method to obtain the energy spectra and the corresponding wave function. By Fourier transforming the position space wave function, the corresponding momentum wave function was obtained for the low‐lying states corresponding to the ground and first excited states. The positions and momentum of Shannon entropy and Fisher Information entropies were calculated numerically. Finally, the Bialynicki‐Birula and Mycielski and the Stam‐Cramer‐Rao inequalities for the Shannon entropy and Fisher Information entropies, respectively, were tested and were found to be satisfied for all cases considered.
In this paper, a three parameter (α, λ, μ) ‘Hartree and Ingman (1933)’ type correlated wave function has been used to derive an analytical model to quantify the two electron systems. The model is used to construct the expressions for the single particle wave functions [ 𝜓(𝑟) , 𝜙(𝑝)] in coordinate space and momentum space respectively with the subscripts marked as correlated (c) and uncorrelated (uc) ones. Further, the correlated and uncorrelated wave functions are used to form the single particle charge densities [𝜌(𝑟) , 𝛾(𝑝)] in coordinate space and momentum space. These wave functions and the single particle charge densities are used to compute the numerical values of the Shannon (S) and Fisher (I) information entropies for both the correlated and uncorrelated systems in coordinate space and momentum spaces and furthermore to examine their correlation effects in coordinate and momentum spaces. The uncorrelated and correlated values of Shannon information entropies indicate the reciprocity between the coordinate and momentum spaces such that low values of the Shannon information entropy in coordinate space ( 𝑆𝜌), are associated with the high values of Shannon information entropy in momentum space (𝑆𝛾) . The uncorrelated and correlated values of Shannon information entropies (S) also validate the entropic uncertainty relations Bialynicki-Birula and Mycielski (BBM) inequality which reads as (𝑆𝜌+𝑆𝛾)≥3(1+𝑙𝑛𝜋), as well as another stronger version of the uncertainty relation that is (𝑆𝜌𝑐+𝑆𝛾𝑐)>(𝑆𝜌𝑢𝑐+𝑆𝛾𝑢𝑐). Further, both the products of uncorrelated and correlated values of Fisher information entropies 𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 and 𝐼𝜌𝑐𝐼𝛾𝑐 satisfy the inequality condition 𝐼𝜌𝑐𝐼𝛾𝑐>𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 in coordinate and momentum spaces. At the same time these products also satisfy the Fisher based uncertainty relation, 𝐼𝜌𝐼𝛾≥36. The consistency of the results satisfying the uncertainty relations can be checked from the data demonstrated in the Table 1 and Table 2.