Density of a many-fermion system in a harmonic-oscillator potential

Abstract

(ricevuto il 17 Dicembre 1982) PACS. 21.10.-General and average properties of nuclei; properties of nuclear energy levels.Summary. -The ground-state density of a system of noninteraeting fermions in a harmonic-oscillator potential well can be given in terms of the wave functions of a few single-particle states around the Fermi level.We are going to demonstrate that the derivative of the density of a system of fermions occupying the lowest levels in a three-dimensional harmonic-oscillator well can be written in terms of a few radial wave functions. If N major oscillator shells are occupied, the derivative implies only the last occupied shell and the first nonoceupied one.In the one-dimensional case, the relation is very simple. If _F is the quantum number of the last occupied shell (Fermi level), and if the qi's are the normalized wave functions, one has In this relation, ~ is the density and a o is the harmonic-oscillator length~o being the oscillator frequency. The proof of relation (1) is given in ref.(1) and is based on recurrence relations fulfilled by the harmonic-oscillator wave functions. The method cannot be extended to the three-dimensional case easily. We therefore prefer to start Q) J. CUGNON and O.