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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Quantum information measures in the confined Robin harmonic oscillator: uncertainty, Shannon entropy and Onicescu energy

Dehua Wang, Tiantian Tang, Binhua Chu

Published Feb 2, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1080/14786435.2026.2622159

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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.
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.
Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ( x c ). In the symmetric case, a wide range of x c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x c , all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. image
In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator. Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology,etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and chemical changes. The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly, in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained situations. The most prominent amongst these are: Fisher information, Shannon entropy, Renyi entropy , Tsallis entropy, Onicescu energy and several complexities. Arguably, these are the most effective measures of uncertainty, as they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several physico-chemical properties of a system under investigation. Kullback-Leibler divergence or relative entropy describes how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination between two states. In other words, it extracts the change of information in going from one state to another.
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 Information‐based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) are employed to understand the influence of radial confinement in an isotropic harmonic oscillator (IHO). The transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (). For a given quantum state, accurate results are provided by applying the respective exact analytical wave function in the r‐space. The p‐space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments () as in r‐ and p‐spaces. A detailed, systematic analysis is performed for a 3D confined harmonic oscillator (CHO) with respect to state indices , and . It has been found that the CHO acts as a bridge between particles in a spherical box and the 3D free IHO. In the low‐ region, increases while decrease with the rise in . At moderate , there exists an interaction between two competing factors: i) radial confinement (localization) and ii) accumulation of radial nodes with growth of (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.
The study of quantum information entropy in confined quantum systems plays a pivotal role in quantum information science. In this work, we investigate the quantum information entropy of a particle in a quantum well under an external electric field by calculating the Rényi entropy (), Tsallis entropy () and the Onicescu information energy (). Our findings reveal that these entropic measures exhibit a strong dependence on the entropy order ( α ). Although both and demonstrate a decreasing trend with increasing α for any given quantum state, they differ significantly in their sensitivity to α variations, displaying distinct behaviors that provide unique insights into quantum uncertainty. For the Onicescu information energy (), a sharp transition around α = 1 appears. This characteristic could be significant for designing and analyzing quantum systems, particularly in enhancing their information processing capabilities. By further examining how these entropic measures vary with quantum well width and electric field strength, we gain a deeper understanding of the system's controllability. This empowers researchers to select optimal metrics for specific applications and improve the accuracy of quantum analyses. This study underscores the importance of entropy‐based analysis in quantum mechanics for advancing quantum control, information theory, and technology development.