Confinement of an electron in an image potential and an external electrostatic field

The quasi-classical approximation is used to determine the positions of the classical turning points upon motion of an electron that is bound by an image field and a constant homogeneous electric field of the same direction. Power expansions of the coordinates of the turning points in a wide range of electron energies and field strengths are obtained. The mechanism of one-dimensional confinement of an electron, which determines a completely discrete spectrum of states, is described. The dependence of the spatial width of the confinement region on the field strength and electron energy is determined. The dependences of the electron energy in different states on the external field strength are calculated numerically. Quasi-classical quantization is performed, and the dependence of the electron energy on the width of the confinement region is determined. The energy interval of a maximum density of electron states is found, which is determined by the dependence of the width of the confinement region on the electric field strength.