  In this paper we focus on energy flows in simple quantum systems. This is
achieved by concentrating on the quantum Hamilton-Jacobi equation. We show how
this equation appears in the standard quantum formalism in essentially three
different but related ways, from the standard Schr\"{o}dingier equation, from
Lagrangian field theory and from the von Neumann-Moyal algebra. This equation
allows us to track the energy flow using the energy-momentum tensor, the
components of which are related to weak values of the four-momentum operator.
This opens up a new way to explore these components empirically. The algebraic
approach enables us to discuss the physical significance of the underlying
non-commutative symplectic geometry, raising questions as to the structure of
particles in quantum systems.
