Universal trend of the information entropy of a fermion in a mean field

Abstract

We calculate the information entropy of single-particle states in position-space $S_r$ and momentum-space $S_k$ for a nucleon in a nucleus, a $\Lambda$ particle in a hypernucleus and an electron in an atomic cluster. It is seen that $S_r$ and $S_k$ obey the same approximate functional form as functions of the number of particles, $S_r$ ({\rm or} $S_k)=a+b{N}^{1/3}$ in all of the above many-body systems in position- and momentum- space separately. The net information content $S_r+S_k$ is a slowly varying function of $N$ of the same form as above. The entropy sum $S_r+S_k$ is invariant to uniform scaling of coordinates and a characteristic of the single-particle states of a specific system. The order of single-particle states according to $S_r+S_k$ is the same as their classification according to energy keeping the quantum number $n$ constant. The spin-orbit splitting is reproduced correctly. It is also seen that $S_r+S_k$ enhances with excitation of a fermion in a quantum-mechanical system. Finally, we establish a relationship of $S_r+S_k$ with the energy of the corresponding single-particle state i.e. $S_r+S_k=k\ln (\mu E+\nu )$. This relation holds for all the systems under consideration.

Comments: 9 pages, latex, 6 figures