Quantum geodesics in quantum mechanics

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus Omega 1 depending on the Hamiltonian p(2)/2m+V(x), and a flat quantum connection del with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or `geodesics') is exactly Schr & ouml;dinger's equation. The connection del preserves a generalised `skew metric' given by the canonical symplectic structure lifted to a certain rank (0,2) tensor on the extended phase space where we adjoin a time variable. We also apply the same approach to the Klein Gordon equation on Minkowski spacetime with a background electromagnetic field, formulating quantum `geodesics' on the relativistic Heisenberg algebra with proper time for the external geodesic parameter. Examples include a relativistic free particle wave packet and a hydrogen-like atom.