Quantum Uncertainty Relations for Thermodynamic Energy Flows

The Heisenberg uncertainty relation, which links the uncertainties of the position and momentum of a particle, has an important footprint on the quantum behavior of a physical system. Analogous to this principle, we propose that thermodynamic currents associated with work, heat, and internal energy satisfy their own uncertainty relations. To formalize this idea, we represent these currents by well-defined Hermitian operators, constructed so that their expectation values match the corresponding average currents. Because these operators generally do not commute, the resulting quantum currents differ fundamentally from their classical counterparts. Using the Robertson-Schr\"odinger uncertainty relation, we derive various uncertainty relations that link different thermodynamic flows. We further illustrate this approach by applying it to quantum batteries, where we derive an energy-power uncertainty relationship and show how measurements affect the fluctuations.