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

摘要

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 ShannonS, Renyi, and TsallisT(alpha)entropies; Onicescu energiesO; and Fisher informationI. It is shown that a transition to the 2D geometry lifts the 1D degeneracy of theR(alpha)position componentsS(rho),O-rho, andR(rho)(alpha). Among many other findings, it is established that the lower limit alpha(TH)of the semi-infinite range of the dimensionless Renyi/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 are1/4and1/2, 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 alpha THNis greater than one half, its Renyi uncertainty relation for the sum of the position and wave vector componentsR rho alpha+R gamma alpha 2 alpha-1is 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 alpha not smaller than one half. For both configurations, the lowest-energy level at alpha= 1/2does saturate Renyi and Tsallis entropic inequalities. Other properties are discussed and analyzed from mathematical and physical points of view.