  In this paper we extend the WKB-like `non-relativistic' expansion of the
minimally coupled Klein--Gordon equation after Kiefer and Singh [1],
L\"ammerzahl [2] and Giulini and Gro{\ss}ardt [3] to arbitrary order in
$c^{-1}$, leading to Schr\"odinger equations describing a quantum particle in a
general gravitational field, and compare the results with canonical
quantisation of a free particle in curved spacetime, following Wajima et al.
[4]. Furthermore, using a more operator-algebraic approach, the Klein--Gordon
equation and the canonical quantisation method are shown to lead to the same
results for some special terms in the Hamiltonian describing a single particle
in a general stationary spacetime, without any `non-relativistic' expansion.
