Curl-free vector potential observation on the macro-scale and a new dimension to the Lorentz trajectory

The recently reported curl-free vector potential observation on the macro-scale, which has been attributed to a macro-scale matter wave associated with the quantum modulation of the de Broglie wave along the magnetic field, is seen to be in apparent contravention with the Lorentz equation of classical electrodynamics, which as a descriptor on the macro-scale takes no cognizance of a curl-free vector potential. The two different formalisms-a deterministic one a la Lorentz and a probabilistic one by the quantum modulation-both existing on the macro-scale and, though in apparent contravention, complement each other and are essential for a complete description of all phenomena relating to charged particle dynamics. The quantum modulations which arise in consequence of transition across Landau levels appear as 'hole' auto-excitations in the Lorentz trajectory involving internal inelastic scattering. This lends a new dimension to the Lorentz trajectory which, though on the classical macro-scale, acts like a 'quantum resonator' and can now exist in a number of self-excited states labelled by the Landau level interval involved in the excitation, and which are of quantum origin and on the macro-scale, with the quantum modulations being the excitations.