Electron in a quantum well with charges on the walls

An electron is considered in a one-dimensional quantum well, on the walls of which screened positive charges are located. By numerically solving the boundary value problem for the Schrödinger equation, the wave functions and eigenvalues of the electron energy are found. It is shown that both positive and negative discrete values of the electron energy are possible for various combinations of the quantum well width and screening radius. Due to size quantization and Coulomb interaction, zero values of the electron momentum are possible, which is associated with Fermi — Dirac condensation.