Classical integrability and quantum velocity fields

We study a stationary velocity field, defined via the probability current density for an energy eigenstate of a quantum system, and its classical analog determined by the Hamilton-Jacobi equation. For classically integrable systems, the quantum-classical correspondence can be properly established by specifying constants of motion in addition to the Hamiltonian. For classically chaotic systems, the correspondence is broken down; the manifestation of classical chaos in the quantum domain is shown numerically to be the appearance of turbulence in the quantum velocity field.