  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.
