An infinite system of electrons in its ground state, subject to a very slowly varying external potential, has slowly varying properties, and can be described by a gradient expansion theory. However, when, in addition, either (1) there is a spatially rapidly varying perturbing potential, or (2) there are regions in which the electron density drops to zero (e.g., electrons in an oscillator potential), the density and other properties of the system exhibit additional spatial oscillations, which we call quantum oscillations. An example are the Friedel oscillations in metals. In the present paper we develop a general theory of these oscillations for one-dimensional, noninteracting electrons. Illustrations are worked out which show that when the quantum oscillations are superposed on the smooth "semiclassical" results, one can obtain very accurate approximations to the exact densities.

• Received 30 October 1964

DOI:https://doi.org/10.1103/PhysRev.137.A1697