Circle-valued phase and the Wallstrom quantization condition in Madelung hydrodynamics

Abstract

Madelung showed that the Schrodinger equation can be rewritten, away from the zeros of the wave function, as hydrodynamic equations for a density and a velocity potential. Wallstrom [Phys. Rev. A 49, 1613 (1994)] showed that the rewriting is not an equivalence: the hydrodynamic system admits solutions whose circulation around nodes is an arbitrary real multiple of h/m, and equivalence requires imposing circulation quantization loop by loop, by hand. We examine a single kinematic postulate, the compact-phase postulate: the phase is fundamentally circle-valued, and the circle has action-circumference h. We prove three statements. First, a continuous circle-valued phase field winds an integer number of times around every loop in the node complement, so the fractional-circulation solutions are excluded kinematically. Second, compactness alone quantizes circulation in an arbitrary unit; the postulated circumference fixes the unit to h/m, and the circumference h is displayed independently by measured ladders in superconducting rings (h/2e) and in superfluid helium (h/m), and by the Aharonov-Bohm period (h/e), all in coherent-condensate systems. Third, under an integer-free regularity condition at the nodes, the postulate together with the Madelung equations is equivalent to exact scalar quantum mechanics at strong-solution (H1_loc) grade. We state the caveat explicitly: over the node complement, given continuity, the circle-valued phase is logically equivalent to the quantization condition it subsumes. The gain is at the level of axioms: one universal kinematic postulate with a calibrated, independently measured unit in place of a per-solution family of integral constraints.