Local energy and power for 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 transferred by an external electrical field to a many-particle system of interacting spinless electrons. In analogy with the definition of the (local) presence and current probability densities, 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, such intriguing features are only reflected in the expression for the local power when the volume $\Omega$ is finite. Otherwise, for closed systems with $\Omega \to \infty$, we recover a classical-like single-particle expression.