  The neutron-rich even-even nuclei $^{26-40}$Mg, $^{28-46}$Si, $^{30-48}$S,
and $^{32-56}$Ar are calculated with the RMF model and the phase-shift electron
scattering method. Results show that level inversion of the 2$s_{1/2}$ and
1$d_{3/2}$ proton states may occur for the magnesium, silicon, sulphur, and
argon isotopes with more neutrons away from the stability line. Calculations
show that the variation of the central charge densities for $^{30-48}$S, and
$^{32-56}$Ar are very sensitive to the 2$s_{1/2}$ and 1$d_{3/2}$ proton state
level inversion, and the level inversion can lead to a large measurable central
charge depletion to the charge density distributions for the neutron-rich
isotopes. Calculations also show that the charge density differences between
the isotopes with and without central charge depletion can reveal not only the
level inversion of the 2$s_{1/2}$ and 1$d_{3/2}$ proton states but also the
behavior of the proton wave functions of both states. The results can provide
references for the possible study of the nuclear level inversion and nuclear
bubble phenomenon with electron scattering off short-lived nuclei at RIKEN
or/and GSI in the future. In addition, direct nuclear reaction $^{44}$S(n,
d)$^{43}$P or $^{44}$S($^{3}$H, $\alpha$)$^{43}$P might also be a possible way
to study the 2$s_{1/2}$ and 1$d_{3/2}$ proton state level inversion.
