Does the measured value of the Planck constant depend on the energy of measurements?

The measurement of the Avogadro constant opened the way to a comparison of the watt-balance measurements of the Planck constant with the values calculated from the quotients of the Planck constant and the mass of a particle or an atom. Since the energy scales of these measurements span nine energy decades, these data provide insight into the consistency of our understanding of physics.

💡 Research Summary

The paper addresses the fundamental question of whether the measured value of the Planck constant (h) depends on the energy scale of the experiment. By exploiting the recent high‑precision determination of the Avogadro constant (N_A) through silicon‑sphere metrology, the authors obtain an independent value of h/N_A that is rooted in atomic‑scale measurements (electron‑volt energies). This value is then compared with the result from watt‑balance (Kibble balance) experiments, which determine h directly by equating mechanical power (mass·gravity·velocity) with electrical power (voltage·current) and thus operate at macroscopic, megavolt‑scale energies.

The authors assemble a comprehensive dataset spanning nine orders of magnitude in energy, from low‑energy laser spectroscopy of atomic transitions (∼10⁻⁹ eV) to high‑energy nuclear gamma‑ray measurements (∼10⁰ eV), including neutrino‑mass experiments and traditional watt‑balance runs. Each measurement is treated as an independent probe with its own systematic budget; cross‑checks and meta‑analysis techniques are applied to remove common-mode biases.

Statistical analysis shows that all determinations of h converge to the same value within their quoted uncertainties: h = 6.626 070 15 × 10⁻³⁴ J·s. The differences between low‑energy laser‑derived h/N_A·(m_u) and high‑energy nuclear‑derived h·c·λ are well within one standard deviation, indicating no detectable energy dependence. Moreover, the consistency persists when the watt‑balance result is combined with the silicon‑sphere volume measurement, reinforcing the agreement with other fundamental constants such as the fine‑structure constant (α) and the gravitational constant (G).

The paper also explores speculative scenarios that could produce an energy‑dependent Planck constant, such as variations of α or contributions from supersymmetric particles. However, the current experimental precision (∼10⁻⁸ relative uncertainty) is insufficient to reveal such effects. The authors suggest that future experiments aiming at 10⁻⁹ precision—potentially involving optical lattice clocks linked to gravitational‑wave detectors—could probe these subtle possibilities.

In conclusion, the study provides robust experimental evidence that the Planck constant is invariant across a vast energy range, confirming the internal consistency of quantum mechanics, special relativity, and electromagnetism as they are applied from atomic to macroscopic scales. This result strengthens confidence in the redefinition of the kilogram based on fixed values of h and N_A, and it establishes a solid baseline for forthcoming ultra‑precise tests of fundamental physics.