Theory of the Robin quantum wall in a linear potential. I. Energy spectrum, polarization and quantum-information measures

Information-theoretical concepts are employed for the analysis of the interplay between a transverse electric field $\mathscr{E}$ applied to a one-dimensional surface and Robin boundary condition (BC), which with the help of the extrapolation length $\Lambda$ 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 Schr\"{o}dinger equation are derived and used for calculating energies, dipole moments, position $S_x$ and momentum $S_k$ quantum information entropies and their Fisher information $I_x$ and $I_k$ and Onicescu information energies $O_x$ and $O_k$ counterparts. It is shown that the weak (strong) electric field changes the Robin wall into the Dirichlet, $\Lambda=0$ (Neumann, $\Lambda=\infty$), 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 $S_x+S_k$. Analogously, at $\Lambda=0$ and $\Lambda=\infty$ the position and momentum Fisher informations (Onicescu energies) depend on the applied voltage as $\mathscr{E}^{2/3}$ ($\mathscr{E}^{1/3}$) and its inverse, respectively, leading to the field-independent product $I_xI_k$ ($O_xO_k$). Peculiarities of their transformations at the finite nonzero $\Lambda$ 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.