The quantum mechanical wavefunction of a freely falling hydrogen atom will remain unchanged
to �first order in the presence of sufficiently weak tidal gravitational fields, giving rise to a form
of incompressibility not seen in classical systems. The fact that the electron is in a bound state
and that only discrete energy level transitions can occur causes the atom to behave differently than
a classical object in an inhomogeneous gravitational field under certain circumstances. A time-
dependent energy shift that is overlooked in a widely-used gauge choice will be explored, and shown
to be a real physical effect through gauge invariance. Quantum incompressibility of the wavefunction
governing the electron super�fluid of a Type-I superconductor will also lead to non-classical behavior
in the presence of tidal gravitational �fields, via coupling between gravitational and electromagnetic
�fields. Theoretical and experimental implications will be discussed
