Semiclassical model of a one-dimensional quantum dot

A one-dimensional quantum dot at zero temperature is used as an example for developing a consistent semiclassical method. The method can also be applied to systems of higher dimension that admit separation of variables. For electrons confined by a quartic potential, the Thomas-Fermi approximation is used to calculate the self-consistent potential, the electron density distribution, and the total energy as a function of the electron number and the effective electron charge representing the strength of interaction between electrons. Use is made of scaling with respect to the electron number. An energy quantization condition is derived. The oscillating part of the electron density and both gradient and shell corrections to the total electron energy are calculated by using the results based on the Thomas-Fermi model and analytical expressions derived in this study. The dependence of the shell correction on the interaction strength is examined. Comparisons with results calculated by the density functional method are presented. The relationship between the results obtained and the Strutinsky correction is discussed.