  We consider quantum mechanics on constrained surfaces which have
non-Euclidean metrics and variable Gaussian curvature. The old controversy
about the ambiguities involving terms in the Hamiltonian of order hbar^2
multiplying the Gaussian curvature is addressed. We set out to clarify the
matter by considering constraints to be the limits of large restoring forces as
the constraint coordinates deviate from their constrained values. We find
additional ambiguous terms of order hbar^2 involving freedom in the
constraining potentials, demonstrating that the classical constrained
Hamiltonian or Lagrangian cannot uniquely specify the quantization: the
ambiguity of directly quantizing a constrained system is inherently
unresolvable. However, there is never any problem with a physical quantum
system, which cannot have infinite constraint forces and always fluctuates
around the mean constraint values. The issue is addressed from the perspectives
of adiabatic approximations in quantum mechanics, Feynman path integrals, and
semiclassically in terms of adiabatic actions.
