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

TL;DR

This paper develops a quantum framework for analyzing local energy and power transfer in many-electron systems under external electric fields, revealing quantum-specific non-local effects and classical-like behaviors in finite volumes.

Contribution

It introduces a local energy operator and derives expressions for local power, bridging quantum and classical descriptions in many-particle systems with spatial resolution.

Findings

Quantum local power exhibits non-local sources and sinks.

Finite volume analysis reveals quantum features absent in infinite systems.

Classical-like effects emerge from current-force correlations and boundary energy flows.

Abstract

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.