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

TL;DR

This paper analytically investigates how a transverse electric field affects a Robin quantum wall, analyzing energy spectra, polarization, and quantum-information measures, revealing field-induced boundary condition transformations and invariances in information sums and products.

Contribution

It provides exact solutions for the Schrödinger equation with Robin boundary conditions under an electric field, exploring their impact on quantum-information measures and boundary condition transformations.

Findings

Electric field transforms Robin boundary conditions into Dirichlet or Neumann types.

Quantum information entropies exhibit logarithmic dependence on electric field intensity.

Certain information measure products remain invariant under varying electric fields.

Abstract

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.