Quantum information measures of the one-dimensional Robin quantum well

Abstract

Shannon quantum information entropies $S_{x,k}$, Fisher informations $I_{x,k}$, Onicescu energies $O_{x,k}$ and statistical complexities $e^{S_{x,k}}O_{x,k}$ are calculated both in the position (subscript $x$) and momentum ($k$) representations for the Robin quantum well characterized by the extrapolation lengths $\Lambda_-$ and $\Lambda_+$ 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 $\Lambda$ for the symmetric, $\Lambda_-=\Lambda_+=\Lambda$, and antisymmetric, $\Lambda_-=-\Lambda_+=\Lambda$, 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 $\Lambda$. It is confirmed, in particular, that the entropic uncertainty relation $S_{x_n}+S_{k_n}\geq1+\ln\pi$ and general inequality $e^SO\geq1$, 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 $S_{x_{n+1}}(\Lambda)+S_{k_{n+1}}(\Lambda)\geq S_{x_n}(\Lambda)+S_{k_n}(\Lambda)$ that is correct for the Neumann ($\Lambda=\infty$) or Dirichlet ($\Lambda=0$) boundary conditions, is violated. Other analytic and numerical results for all measures are discussed too and their physical meaning is highlighted.