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Physica Scripta

Information entropy, fractional revivals and Schrödinger equation with position-dependent mass

Shahid Iqbal

Published Dec 23, 2021• 5 citations
Publisher

Abstract

Abstract Information entropy has played a key role in a wide range of disciplines, for instance, classical and quantum information processing, quantum computing, quantum dynamics and quantum metrology. Here, we develop an information theoretic formalism using Shannon entropy, to investigate the quantum dynamics of Hamiltonian systems with position-dependent mass. Such systems are of fundamental interest in many areas, for instance, condensed matter, mathematical physics and foundations of quantum mechanics. We explore the phenomenon of fractional revivals for the temporal evolution of wave-packet solutions of Schrödinger equation with position-dependent mass by studying, analytically and numerically, the time-development of Shannon information entropy in position and momentum spaces. It is shown by our numerical results that the effect of spatially varying mass on the fractional revivals can not be fully harnessed using conventional measures, for instance, autocorrelation function. However, based on our numerical analysis it is concluded that information entropy is not only more sensitive to identify the fractional revivals but it also better elucidates the effect of position-dependent mass on the structure of fractional revivals in the form of symmetry breaking.

DOI
10.1088/1402-4896/ac4633

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Abstract Information entropy has played a key role in a wide range of disciplines, for instance, classical and quantum information processing, quantum computing, quantum dynamics and quantum metrology. Here, we develop an information theoretic formalism using Shannon entropy, to investigate the quantum dynamics of Hamiltonian systems with position-dependent mass. Such systems are of fundamental interest in many areas, for instance, condensed matter, mathematical physics and foundations of quantum mechanics. We explore the phenomenon of fractional revivals for the temporal evolution of wave-packet solutions of Schrödinger equation with position-dependent mass by studying, analytically and numerically, the time-development of Shannon information entropy in position and momentum spaces. It is shown by our numerical results that the effect of spatially varying mass on the fractional revivals can not be fully harnessed using conventional measures, for instance, autocorrelation function. However, based on our numerical analysis it is concluded that information entropy is not only more sensitive to identify the fractional revivals but it also better elucidates the effect of position-dependent mass on the structure of fractional revivals in the form of symmetry breaking.
Abstract Position-dependent-mass systems are of great importance in many physical situations, such as the transport of charge carriers in semiconductors with non-uniform composition and in the theory of many-body interactions in condensed matter. Here we investigate, numerically and analytically, the phenomenon of fractional revivals in such systems, which is a generic characteristic manifested by the wave-packet evolution in bounded Hamiltonian systems. Identifying the fractional revivals using specific probes is an important task in the theory of quantum measurement and sensing. We numerically simulate the temporal evolution of probability density and information entropy density, which manifest self-similarly recurring interference patterns, namely, quantum carpets. Our numerical results show that the quantum carpets not only serve as an effective probe for recognizing the fractional revivals of various order but they efficiently describe the effect of spatially-varying mass on the structure of fractional revivals, which is manifested as a symmetry breaking in their designs.
Wave-packet fractional revivals is a relevant feature in the long time-scale evolution of a wide range of physical systems, including atoms, molecules, and nonlinear systems. We show that the sum of information entropies in both position and momentum conjugate spaces is an indicator of fractional revivals by analyzing three different model systems: (i) the infinite square well, (ii) a particle bouncing vertically against a wall in a gravitational field, and (iii) the vibrational dynamics of hydrogen iodide molecules. This description in terms of information entropies complements the usual one in terms of the autocorrelation function.
In this paper, we analyze quantum information measures in fractional quantum mechanics using the Riemann-Liouville derivative formalism adopted here. In this case, we initially reconsider the conventional definitions of Shannon entropy and Fisher information, subsequently extending them to fractional quantum systems described by nonlocal differential operator frameworks adopted. Within this generalized formulation, fractional expressions of Shannon entropy and Fisher information are constructed and their mathematical structures examined thoroughly. Also, the formalism is then applied to the quantum harmonic oscillator, yielding explicit analytical expressions derived as functions of the fractional parameter therein. The obtained results demonstrate that fractional derivatives alter the localization properties of probability densities and generate nontrivial variations in information content and sensitivity across system behavior. In this context, the fractional parameter plays a central role in controlling deviations from the standard quantum information measures framework. Also, the study establishes a consistent framework for describing information-theoretic properties of quantum systems governed by nonlocal dynamics.
ABSTRACT We investigate the quantum information entropy for a class of single hyperbolic potentials within the context of the fractional Schrödinger equation (FSE). We find that as the derivative variable decreases, the position entropy density function becomes more localized, and its peak heightens. However, there are differences in the degree of localization between the position entropy density functions for each hyperbolic potential, which can be attributed to the varying sizes of the potentials. Conversely, in momentum space, the momentum entropy density function becomes more delocalized, and its peak lowers as the derivative variable decreases for both hyperbolic potentials studied. Our analysis also examines the BBM inequality, demonstrating that it is satisfied for different values of the potential depths. Finally, we explore the Fisher entropy and observe that it increases in position space while decreasing in momentum space as the depth of the wells increases. Our findings provide new insights into the behavior of quantum systems governed by hyperbolic potentials within the fractional Schrödinger framework. The observed localization effects in position space, delocalization in momentum space, and the validation of the BBM inequality highlight the role of fractional derivatives in modifying quantum entropy measures. These results deepen our understanding of quantum information entropy in non‐local quantum systems. They may have implications for fields such as quantum transport in disordered media, semiconductor physics, and the study of anomalous diffusion processes in quantum mechanics.
Abstract In this work, we examine the quantum information characteristics of a hydrogen-like system under a Coulomb potential by using previously derived exact solutions of the generalized Schrödinger equation formulated within the conformable fractional framework. The approach introduces a fractional parameter (0<α≤1)(0 < \alpha \le 1)(0<α≤1) ( 0 < α ≤ 1 ) , which permits a continuous transition between standard quantum mechanics and a fractional description. Utilizing these available analytical solutions, we determine the associated energy spectrum and wavefunctions and employ them to quantify the informational structure of the system. In particular, we calculate the Shannon entropy in both position and momentum spaces, along with the Fisher information, as fundamental indicators of uncertainty, spatial spreading, and sensitivity to local variations of the probability density. The results show that the fractional parameter (α)(\alpha )(α) ( α ) produces nontrivial corrections to these quantum information measures, indicating modifications in localization properties and momentum profiles. Also, the formalism recovers the conventional hydrogenic limit when (α→1)(\alpha \rightarrow 1)(α→1) ( α → 1 ) , which validates the consistency of the formulation. In this context, these results show that the conformable fractional framework constitutes a consistent extension for analyzing quantum systems via information-theoretic quantities without modifying the exact solutions, providing a systematic approach to investigate deviations from integer-order dynamics.
Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.