Abstract Shannon quantum information entropies , Fisher informations , Onicescu energies , and statistical complexities are calculated both in the position (subscript x) and momentum (k) representations for the Robin quantum well characterized by the extrapolation lengths and at the two confining surfaces. The analysis concentrates on finding and explaining the most characteristic features of these quantum information measures in the whole range of variation of the Robin distance Λ for the symmetric, , and antisymmetric, , geometries. Analytic results obtained in the limiting cases of the extremely large and very small magnitudes of the extrapolation parameter are corroborated by the exact numerical computations that are extended to the arbitrary length Λ. It is confirmed, in particular, that the entropic uncertainty relation and general inequality , which is valid both in the position and momentum spaces, hold true at any Robin distance and for every quantum state n. For either configuration, there is a range of the extrapolation lengths where the rule that is correct for the Neumann ( ) or Dirichlet ( ) boundary conditions, is violated. Other analytic and numerical results for all measures are discussed too and their physical meaning is highlighted.
Abstract Shannon quantum information entropies , Fisher informations , Onicescu energies , and statistical complexities are calculated both in the position (subscript x) and momentum (k) representations for the Robin quantum well characterized by the extrapolation lengths and at the two confining surfaces. The analysis concentrates on finding and explaining the most characteristic features of these quantum information measures in the whole range of variation of the Robin distance Λ for the symmetric, , and antisymmetric, , geometries. Analytic results obtained in the limiting cases of the extremely large and very small magnitudes of the extrapolation parameter are corroborated by the exact numerical computations that are extended to the arbitrary length Λ. It is confirmed, in particular, that the entropic uncertainty relation and general inequality , which is valid both in the position and momentum spaces, hold true at any Robin distance and for every quantum state n. For either configuration, there is a range of the extrapolation lengths where the rule that is correct for the Neumann ( ) or Dirichlet ( ) boundary conditions, is violated. Other analytic and numerical results for all measures are discussed too and their physical meaning is highlighted.
With the development of quantum technology and information science, significant attention is focused on studying the quantum information properties in confined systems. In this work, we present a systematic investigation of quantum information measures – including position-momentum uncertainty (Δx,Δp), Shannon entropy (Sx, Sp), and Onicescu energy (Ox, Op) in a confined Robin harmonic oscillator (CRHO) with symmetric boundary conditions λL = λR = λ. It is interesting to find that the Heisenberg uncertainty relation ΔxΔp≥ℏ/2, which holds under the Dirichlet boundary condition (λ = 0), is violated for the ground state of the CRHO. In contrast, the entropic uncertainty relation (Sx + Sp ≥ 1 + lnπ) remains satisfied at any Robin length λ and quantum state, demonstrating its fundamental advantage and broader applicability in quantum systems. For a fixed Robin length, we observe that as the quantum well width increases within a finite range, the position-space Shannon entropy Sₓ increases, while the Onicescu energy Oₓ decreases. Despite their opposing tendencies, both Shannon entropy and Onicescu energy reflect the same physical insight regarding the localisation-delocalisation property for the CRHO system. In addition, the dependence of these information measures on the Robin length is discussed, and their physical implications are highlighted. This study demonstrates that boundary conditions play a critical role in quantum information metrics, thereby informing the design and optimisation of quantum devices such as quantum dots, nanoscale resonators and atom-based sensors.
Information‐theoretical concepts are employed for the analysis of the interplay between a transverse electric field applied to a one‐dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length Λ zeroes at the interface a linear combination of the quantum mechanical wave function and its spatial derivative, and its influence on the properties of the structure. For doing this, exact analytical solutions of the corresponding Schrödinger equation are derived and used for calculating energies, dipole moments, position and momentum quantum information entropies and their Fisher information and and Onicescu information energies and counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, (Neumann, ), surface. This transformation of the energy spectrum and associated waveforms in the growing field defines an evolution of the quantum‐information measures; for example, it is proved that for the Dirichlet and Neumann BCs the position (momentum) quantum information entropy varies as a positive (negative) natural logarithm of the electric intensity what results in their field‐independent sum . Analogously, at and the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as ( ) and its inverse, respectively, leading to the field‐independent product ( ). Peculiarities of their transformations at the finite nonzero Λ are discussed and similarities and differences between the three quantum‐information measures in the electric field are highlighted with the special attention being paid to the configuration with the negative extrapolation length. image
Information entropic measures such as Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy of a symmetric double‐well potential are calculated in both position and momentum space. Eigenvalues and eigenvectors of this system are obtained through a variation‐induced exact diagonalization procedure. The information entropy‐based uncertainty relation is shown to be a better measure than conventional uncertainty product in interpreting purely quantum mechanical phenomena, such as, tunneling and quantum confinement in this case. Additionally, the phase‐space description provides a semi‐classical explanation for this feature. Total information entropy and phase‐space area show similar behavior with increasing barrier height. image
Abstract Analytic representation of both position and momentum waveforms of the two‐dimensional (2D) circular quantum dots with the Dirichlet and Neumann boundary conditions (BCs) allowed an efficient computation in either space of Shannon S, Rényi, and Tsallis T(α) entropies; Onicescu energies O; and Fisher information I. It is shown that a transition to the 2D geometry lifts the 1D degeneracy of the R(α) position components Sρ, Oρ, and Rρ(α). Among many other findings, it is established that the lower limit αTH of the semi‐infinite range of the dimensionless Rényi/Tsallis coefficient, where one‐parameter momentum entropies exist, is equal to 2/5 for the Dirichlet requirement and 2/3 for the Neumann one. As their 1D counterparts are 1/4 and 1/2, respectively, this simultaneously reveals that this critical value crucially depends not only on the position BC but the dimensionality of the structure too. As the 2D Neumann threshold is greater than one half, its Rényi uncertainty relation for the sum of the position and wave vector components is valid in the range [1/2, 2) only with its logarithmic divergence at the right edge, whereas for all other systems, it is defined at any coefficient α not smaller than one half. For both configurations, the lowest‐energy level at α = 1/2 does saturate Rényi and Tsallis entropic inequalities. Other properties are discussed and analyzed from mathematical and physical points of view.
ABSTRACT Engineering boundary conditions in low‐dimensional structures provides a simple yet powerful way of shaping how quantum information is stored and transported. We investigate a flat 2D Dirichlet waveguide containing one or two finite Neumann windows and compute the bound states in both position and momentum space as functions of the window length. From these wavefunctions, we evaluate three complementary quantum‐information measures ‐Shannon entropy, Onicescu informational energy, and Fisher information in real and reciprocal space, and we analyze the associated entropic and Fisher uncertainty relations. Special attention is paid to the behavior near the critical window lengths at which excited bound states emerge from the continuum, where we derive analytic limiting expressions for the wavefunctions and all information measures. In this regime, we show that the particle effectively inherits the properties of the ground state of a 1D Dirichlet quantum well, independently of the detailed window boundary condition. A systematic comparison between single‐ and double‐window geometries highlights how boundary engineering can be used to tailor localization, sensitivity, and information‐theoretic characteristics of waveguide modes relevant for quantum transport, waveguide‐based qubits, and quantum sensing architectures.
Localization of a particle in the wells of an asymmetric double‐well (DW) potential is investigated here. Information entropy‐based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phase‐space area, are utilized to explain the contrasting effect of localization‐delocalization and role of asymmetric term in such two‐well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi‐degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variation‐induced exact diagonalization method. In brief, information measures and phase‐space analysis can provide valuable insight toward the understanding of such potentials.