How does the complementarity principle affect our understanding of energy conservation in quantum systems, particularly when considering thermal energy at the quantum scale?

The complementarity principle tells us that a quantum system cannot simultaneously reveal its full wave‑like and particle‑like properties, so when we measure its energy we disturb its other observables. Because of this, the energy of a quantum particle can only be known to within an uncertainty that grows with the precision of other measurements, yet the total energy of the system—including its thermal contribution—remains conserved on average. In a thermal quantum system, such as an electron in a nanoscale box, the Heisenberg uncertainty leads to small energy fluctuations, but the expectation value of the energy stays constant over time. For example, if we measure the electron’s energy in a potential well, the wavefunction collapses to a specific energy level while its position becomes highly uncertain, illustrating how complementarity limits simultaneous knowledge. Thus, complementarity reshapes our view of energy conservation by showing that while individual measurements are uncertain, the overall energy, including thermal energy at the quantum scale, is still preserved in expectation.