Abstract Influence of the transverse uniform magnetic field B on position (subscript ρ ) and momentum ( γ ) Shannon quantum-information entropies S ρ , γ Fisher informations I ρ , γ and informational energies O ρ , γ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Comparative analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. A conspicuous distinction between the two spectra are crossings at the increasing induction of the Neumann energies with the same radial quantum number n and adjacent non-positive angular indices m . At the growing B , either system undergoes Landau condensation when its characteristics turn into their uniform field counterparts. It is shown that for the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g. the Dirichlet sum S ρ 00 + S γ 00 on its approach from above to a fundamental limit 2 ( 1 + ln π ) is at any B smaller than the corresponding Neumann quantity what physically means that the former geometry provides more total information about the position and motion of the particle. It is pointed out that the widely accepted disequilibrium uncertainty relation O ρ O γ ≤ 2 π − d , with d being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed; in particular, contrary to the hard-wall QDs, for this configuration the sums S ρ + S γ and the products I ρ I γ and O ρ O γ do not depend on the field. Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.
Abstract Influence of the transverse uniform magnetic field B on position (subscript ρ ) and momentum ( γ ) Shannon quantum-information entropies S ρ , γ Fisher informations I ρ , γ and informational energies O ρ , γ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Comparative analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. A conspicuous distinction between the two spectra are crossings at the increasing induction of the Neumann energies with the same radial quantum number n and adjacent non-positive angular indices m . At the growing B , either system undergoes Landau condensation when its characteristics turn into their uniform field counterparts. It is shown that for the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g. the Dirichlet sum S ρ 00 + S γ 00 on its approach from above to a fundamental limit 2 ( 1 + ln π ) is at any B smaller than the corresponding Neumann quantity what physically means that the former geometry provides more total information about the position and motion of the particle. It is pointed out that the widely accepted disequilibrium uncertainty relation O ρ O γ ≤ 2 π − d , with d being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed; in particular, contrary to the hard-wall QDs, for this configuration the sums S ρ + S γ and the products I ρ I γ and O ρ O γ do not depend on the field. Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.
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.
Analytical solutions of the Schrödinger equation for the one-dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field [Formula: see text] are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small [Formula: see text], the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC-dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two-level approximation elementary explains the negative polarizations as a result of the field-induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann-Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero-voltage position entropies [Formula: see text] are BC independent and for all states but the ground Neumann level (which has [Formula: see text]) are equal to [Formula: see text] while the momentum entropies [Formula: see text] depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC-dependent zero-energy and zero-polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed.
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
The shape complexity of a particle confined in the circular quantum well with an external homogeneous perpendicular magnetic field is investigated. First, the Shannon entropy and the averaging electron probability density, as well as the shape complexities in the position and momentum spaces for several quantum states, are computed and discussed. The quantum confinement effect on the shape complexity of this system is extensively explored. Then, the influence of the magnetic field on the shape complexities of the particle confined in the circular quantum well is discussed. It is demonstrated that the shape complexity exhibits scaled invariance under the influence of the magnetic field. However, when the radius of the quantum well is fixed, the shape complexity of the given quantum state changes with the magnetic field strength. Results suggest that the shape complexity of a particle confined in the quantum well can be changed by varying both the strength of the magnetic field and the size of the well. This study has some practical applications in quantum information measurement of the confined particle and can guide future researches for the disorder characteristics of confined particle in semiconductor quantum dots.
This work provides analytical study of a two-dimensional quantum harmonic oscillator (HO) coupled with a perpendicular magnetic field (B) in the x-direction and an inverse-square potential in the y-direction. The system’s Hamiltonian is exactly solved via variable separation, to obtain the energy levels and wavefunctions composed of Hermite and associated Laguerre polynomials. Fisher information (FI) and Shannon entropy (SE) for both position space (PS) and momentum space (MS) are fully examined under the magnetic field strength and anisotropic potential parameters. The canonical partition function is computed and thermodynamic quantities such as internal energy, entropy, and Gibbs free energy are evaluated. The result shows that the presence of magnetic field enhances spatial localization and reduces uncertainty in PS but has a revised effect in the MS. In the thermodynamic properties, there is asymmetric thermodynamic responses in the x- and y-directions due to the presence of the magnetic field and the inverse-square potential. This study highlights the deep relationship between quantum information theory and thermodynamics in low-dimensional systems with structured interactions.