Two scales of quantum effects in a mesoscopic system of degenerate electrons

A novel nontrivial manifestation of quantum effects in’a spherical mesoscopic system of degenerate electrons (with $N_{\rm e} \lesssim 10^9$) electrons) is demonstrated. Analysis shows that the electron distribution function has a large spatial scale, of the order of the system size, which is much greater than the other spatial scale, the Fermi length of the electron. This result is obtained following four different strategies: an analytic method of semiclassical Green's functions, direct numerical summation of the exact solutions for electron wave functions in an infinitely deep potential well, numerical construction of the exact Green's function using two linearly independent solutions, and the density functional method. The results obtained with these methods are in good qualitative and quantitative agreement with each other. We relate the nature of this phenomenon to quantum shell effects in a spherical mesoscopic system.