  We discuss the internal structure of radially excited charmonium mesons based
on the equal-time and Coulomb gauge Bethe-Salpeter (BS) amplitudes, which are
obtained in lattice QCD. Our simulations are performed with a relativistic
heavy-quark action for the charm quark on the 2+1 flavor PACS-CS gauge
configurations at the lightest pion mass, $M_{\pi}=156(7)$ MeV. The variational
method is applied to the study of optimal charmonium operator for ground and
first excited states of $S$-wave charmonia. We successfully calculate the BS
wave functions of $\eta_c(2S)$ and $\psi(2S)$ states, as well as $\eta_c(1S)$
and $J/\psi$ states, and then estimate the root-mean-square radii of both the
$1S$ and $2S$ charmonium states. We also examine whether a series of the BS
wave functions from the ground state to excited states can be described by a
single set of the spin-independent and spin-dependent interquark potentials
with a unique quark mass. It is found that the quark kinetic mass and, both the
central and spin-spin charmonium potentials, determined from the $2S$ wave
functions, fairly agree with the ones from the $1S$ wave functions. This
strongly supports the validity of the potential description for the charmonium
system, at least, below open-charm threshold.
