Comparative analysis of information measures of the Dirichlet and Neumann two‐dimensional quantum dots

Abstract

AbstractAnalytic 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.