  Ultracold neutrons are great experimental tools to explore the gravitational
interaction in the regime of quantized states. From a theoretical perspective,
starting from a Dirac equation in curved spacetime, we applied a perturbative
scheme to systematically derive the non-relativistic Schr\"odinger equation
that governs the evolution of the neutron's wave function in the Earth's
gravitational field. At the lowest order, this procedure reproduces a
Schr\"odinger system affected by a linear Newtonian potential, but corrections
due to both curvature and relativistic effects are present. Here, we argue that
one should be very careful when going one step further in the perturbative
expansion. Proceeding methodically with the help of the Foldy-Wouthuysen
transformation and a formal post-Newtonian $c^{-2}-$expansion, we derive the
non-relativistic Hamiltonian for a generic static spacetime. By employing Fermi
coordinates within this framework, we calculate the next-to-leading order
corrections to the neutron's energy spectrum. Finally, we evaluate them for
typical experimental configurations, such as that of qBOUNCE, and note that,
while the current precision for observations of ultracold neutrons may not yet
enable to probe them, they could still be relevant in the future or in
alternative circumstances.

General Relativity and Quantum Cosmology;Quantum Physics