  The Hamiltonian of the metric General Relativity derived in our earlier study
(Gravitation, {\bf 17}, 314 - 323 (2011)) is analyzed by the methods of Matrix
Quantum Mechanics. This Hamiltonian is a quadratic function of the momenta
$\pi^{mn}$ conjugate to the spatial components $g_{mn}$ of the metric tensor
$g_{\alpha\beta}$. The Hamiltonian is reduced to the new form which is more
convenient to derive the Schr\"{o}dinger equation for the free gravitational
field. In turn, this Schr\"{o}dinger equation is used to study possible motions
of the free gravitational field(s). In particular, it is shown that harmonic
oscillations of the free gravitational field, or harmonic gravitational waves,
cannot be observed as an actual motion of this field. We also investigate the
internal structure of the solitary gravitational wave.
