Equations of motion for a time-dependent open system: An algebraic approach

Highlights: ► Based on the concept of quantum densities an algebraic equation is introduced. ► Heisenberg equation and hypervirial theorem are derived based on the algebraic equation. ► Quantum Navier–Stokes equation is derived based on the algebraic equation. ► Differential form of the force law and local virial theorem are extracted. ► Central equations in QTAIM are extracted without referring to quantum stationary action. - Abstract: An algebraic approach based on the concept of local densities is introduced in order to provide an alternative derivation of several equations central to the hydrodynamical formulation of quantum mechanics. The origin of this work lays in an algebraic equation which is built based on the concept of quantum densities. This enables us to derive the regional and local forms of several significant quantum laws and equations, namely Heisenberg equation of motion, hypervirial theory and quantum Navier–Stokes equation. In particular, atomic force law and local virial theorem for a time-dependent open system are extracted without referring to rigorous Schwinger's principle of stationary action