Orbital magnetism in RealQM — the charge vortex

A single electron as a real charge density in a 2D harmonic well. A winding state ψ=R(r)eimφ is a real charge vortex: the density circulates as a genuine current j=ρv, carrying angular momentum Lz=m and magnetic moment μz=−mμB. No complex plane is needed for the flow — only the integer m, fixed by single-valuedness (∮v·dℓ=2πm). The field slider drives the exact Fock–Darwin response: linear Zeeman splitting plus the quadratic diamagnetic shift.
Colour = charge density ρ=|ψ|². Dots = charge advected by the current v=(m/r)φ̂. For m≠0 the density is hollow on the axis (the vortex core).
Lz (angular momentum) 0
μz / μB 0
circulation ∮v·dℓ / 2π 0
⟨r²⟩ (mean-square size) 1.000
χdia = −¼⟨r²⟩ −0.250
energy E(m) 1.000
  — Zeeman (−mμBB) 0.000
  — diamagnetic (∝B²) 0.000
split |E(+|m|)−E(−|m|)| 0.000
Atomic units: ℏ=me=e=1, well ω=1, so μB=½ and ωc=B. Levels E=(2nr+|m|+1)Ω−½mB, Ω=√(1+(B/2)²) (Fock–Darwin), with ⟨r²⟩=(2nr+|m|+1)/Ω. χdia=−¼⟨r²⟩ is the 2D Langevin susceptibility — the same ⟨r²⟩-law as the atomic χ=−(NAe²/6mec²)⟨r²⟩.
Fock–Darwin fan: E(m) vs B for m=−2…+2 at the selected nr. The lines split linearly (Zeeman) and bend upward together (diamagnetism). Marker = current B.