Local energy and power in many-particle quantum systems driven by an external electrical field

We derive expressions for the expectation values of the local energy and the local power for a many-particle system of (scalar) charged particles interacting with an external electrical field. In analogy with the definition of the (local) current probability density, we construct a local energy operator such that the time-rate of change of its expectation value provides information on the spatial distribution of power. Results are presented as functions of an arbitrarily small volume Omega and physical insights are discussed by means of the quantum hydrodynamical representation of the wavefunction, which is proven to allow for a clear-cut separation into contributions with and without classical correspondence. Quantum features of the local power are mainly manifested through the presence of non-local sources/sinks of power and through the action of forces with no classical counterpart. Many-particle classical-like effects arise in the form of current-force correlations and through the inflow/outflow of energy across the boundaries of the volume Omega Interestingly, all these intriguing features are only reflected in the expression of the local power when the volume Omega is finite. Otherwise, for closed systems with Omega -> infinity we recover a classical-like single-particle expression.