  We take a closer and new look at the effects of tidal forces on the free fall
of a quantum particle inside a spherically symmetric gravitational field. We
derive the corresponding Schr\"odinger equation for the particle by starting
from the fully relativistic Klein-Gordon equation in order (i) to briefly
discuss the issue of the equivalence principle and (ii) to be able to compare
the relativistic terms in the equation to the tidal-force terms. To the second
order of the nonrelativistic approximation, the resulting Schr\"odinger
equation is that of a simple harmonic oscillator in the horizontal direction
and that of an inverted harmonic oscillator in the vertical direction. Two
methods are used for solving the equation in the vertical direction. The first
method is based on a fixed boundary condition, and yields a discrete-energy
spectrum with a wavefunction that is asymptotic to that of a particle in a
linear gravitational field. The second method is based on time-varying boundary
conditions and yields a quantized-energy spectrum that is decaying in time.
Moving on to a freely-falling reference frame, we derive the corresponding
time-dependent energy spectrum. The effects of tidal forces yield an
expectation value for the Hamiltonian and a relative change in time of a
wavepacket's width that are mass-independent. The equivalence principle, which
we understand here as the empirical equivalence between gravitation and
inertia, is discussed based on these various results. For completeness, we
briefly discuss the consequences expected to be obtained for a Bose-Einstein
condensate or a superfluid in free fall using the nonlinear Gross-Pitaevskii
equation.
