  We reduce Dirac's spinor formalism for a spin 1/2 particle to a complex
wavefunction description in curved spacetimes. We consider a localized
fermionic particle in curved spacetimes and perform an expansion in terms of
the acceleration and curvature around the center of mass of the system,
generalizing the results of [Phys. Rev. D 22, 1922]. Under a non-relativistic
approximation, one obtains a quantum description in a Hilbert space of complex
wavefunctions defined in the rest space of the system. The wavefunction of the
particle then evolves according to a modified Schr\"odinger equation associated
with a symmetric Hamiltonian. When compared to the standard Schr\"odinger
equation for a wavefunction, we obtain corrections in terms of the acceleration
of the system's center of mass and curvature of spacetime along its trajectory.
In summary, we provide a formalism for the use of a complex wavefunction to
describe a localized quantum particle in curved spacetimes.
