Quantum Hall Edge EngineA charged Quantum wave moves inside a strong perpendicular magnetic field, guided by the Hamiltonian H = -1/2 Δ - ΩL_z + 1/2 Ω²r² + V(x,y,t)The term -ΩL_z drives the chiral rotation, and so the edge flow is the angular-momentum part of the magnetic Schrödinger evolution.The density ρ = |ψ|² shows where the wave lives. The smooth threads follow the magnetic probability current j = Im(ψ*∇ψ) - Aρ, with A = Ω(-y, x). Near the boundary, the confining potential bends the wave i