Transitions between degenerate states in a nonrelativistic double-solution framework
Phys. Rev. A 114, 032213 – Published 18 September, 2026
DOI: https://doi.org/10.1103/1l17-tx67
Abstract
Quantizing a single, spinless, nonrelativistic particle constrained to a moving surface, and requiring that Schrödinger's equation is recovered, implies that the surface should be advected by the Madelung velocity field. Moreover, the surface-height gradient, weighted by the squared modulus of the wave function, should have zero net outward flux across closed paths. These constraints determine a phase-coincident companion solution to Schrödinger's equation, which allows us to construct an orthogonal unoccupied state. By supposing that surface fluctuations may induce rare transitions into this state, we motivate an empirical test of de Broglie's double-solution theory. The associated Fermi-golden-rule rate is obtained from first-order perturbation theory in the fluctuation amplitude.
