Quantum-information theory of magnetic field influence on circular dots with different boundary conditions

Abstract: Influence of the transverse uniform magnetic field $\bf B$ on position (subscript $\rho$) and momentum ($\gamma$) Shannon quantum-information entropies $S_{\rho,\gamma}$, Fisher informations $I_{\rho,\gamma}$ and informational energies $O_{\rho,\gamma}$ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. Conspicuous distinction between the 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. For the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g., the Dirichlet sum $S_{\rho_{00}}+S_{\gamma_{00}}$ on its approach from above to a fundamental limit $2(1+\ln\pi)$ 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 $O_\rho O_\gamma\leq(2\pi)^{-\mathtt{d}}$, with $\mathtt{d}$ being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed. 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.