  The Hamiltonian operator describing a quantum particle on a path often
extends holomorphically to a complex neighborhood of the path. When it does, it
can be seen as the local expression of a complex projective structure, and its
perturbations become deformations of that geometric structure. We'll describe
the Hamiltonian of a free particle as a complex projective surface, and we'll
use tools from quasiconformal geometry to study its perturbations. Our main
results are loosely modeled on the algebraic "transformation theory" results
that underpin the exact WKB method. They're meant to serve as a foundation for
efforts to gain a more geometric understanding of the exact WKB method.
