Ultracold Neutron Quantum States above a Polarised Mirror: A Torsion Shift Calculation

Ultracold neutrons bouncing on a horizontal mirror do not fall classically: they occupy quantised gravitational states whose energies — the first levels at 1.41, 2.46, and 3.32 pico-electron-volts — are measured by resonance spectroscopy with a resolution around 1e-03 peV. The neutron wavefunction hovers some fifteen micrometres above the mirror and penetrates it evanescently, which raises a precise question: if the mirror is a magnetised iron film, does the Einstein-Cartan torsion inside the film shift the gravitational levels by acting on the neutron spin during the penetrating fraction of its motion? This paper computes the shift with the wavefunction weight inside the film estimated at 1e-03. Inside the iron the torsion is 6.24e-40 per metre and the spin splitting 1.23e-46 eV, but weighted by the penetration fraction the level shift is 1.23e-37 peV — thirty-four orders of magnitude below the spectroscopic resolution, bounding any multiplier of the torsion coupling at 8.13e+33. The paper also examines the one variant that could in principle do better, a mirror of polarised helium-3 or a ferromagnetic film engineered for deeper wavefunction penetration, and shows the gain is bounded by the penetration physics to a few orders of magnitude. The gravitational quantum states, the most delicate mechanical spectrum in neutron physics, are thereby certified as torsion-free at every relevant level — a useful null for a technique that is a leading candidate to detect other exotic surface forces. Computed from stated assumptions, not measured.