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

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} $S_{k}) = a+bN^{1/3}$ 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.Comment: 9 pages, latex, 6 figure

Application of information entropy to nuclei

Shannon's information entropies in position- and momentum- space and their sum $S$ are calculated for various $s$-$p$ and $s$-$d$ shell nuclei using a correlated one-body density matrix depending on the harmonic oscillator size $b_0$ and the short range correlation parameter $y$ which originates from a Jastrow correlation function. It is found that the information entropy sum for a nucleus depends only on the correlation parameter $y$ through the simple relation $S= s_{0A} + s_{1A} y^{-\lambda_{sA}}$, where $s_{0A}$, $s_{1A}$ and $\lambda_{sA}$ depend on the mass number $A$. A similar approximate expression is also valid for the root mean square radius of the nucleus as function of $y$ leading to an approximate expression which connects $S$ with the root mean square radius. Finally, we propose a method to determine the correlation parameter from the above property of $S$ as well as the linear dependence of $S$ on the logarithm of the number of nucleons.Comment: 10 pages, 10 EPS figures, RevTeX, Phys.Rev.C accepted for publicatio