Quantum geodesics in quantum mechanics

We show that the standard Heisenberg algebra of quantum mechanics admits a noncommutative differential calculus ${\mathrm{\Omega }}^1$ depending on the Hamiltonian $p^2\mathrm{/}2m+V(x)$, and a flat quantum connection $\mathrm{\nabla }$ with torsion such that a previous quantum-geometric formulation of flow along autoparallel curves (or geodesics') is exactly Schr\"odinger's equation. The connection $\nabla$ 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.