‘Number States’ and ‘Pilot Waves’ Hidden in Maxwell’s Classical Equations Available to Purchase

Schrödingers equation with boundary conditions gives quantized energy states for electron waves, but Maxwell’s wave equations have quantized states only by analogies with harmonic oscillators. This problem is addressed by a novel theory of wave‐packets using diffracting Transverse Electric and Transverse Magnetic fields defined by axial H‐ and E‐fields. All transverse fields and gradient operators can together be rotated about the propagation axis at frequencies, independent of the modal frequency. Without altering the axial fields, any helical motion propagates at the group velocity. This is quite different from single frequency helical modes (e.g. Laguerre Gaussian) travelling at the phase velocity. Reversing time and frequency, allows counter rotating helical solutions. These are referred to as adjoint or a fields that may interact and propagate with the classical causal reference or r fields. Overlapping and counter rotating r and a fields with slightly different frequencies interfere, leaving circular polarization states unaltered and creating a nodal structure in the transverse fields distinct from the nodal structure in the axial fields. Number states arise from requiring that transverse and axial nodes co‐locate with integral spacings to form a wave‐packet,. The a fields act as pilot waves for future potential positions of a quantized interaction between r and a fields. Uncertainty in the position of the overlap leads to conventional probabilistic quantum interpretations. The a fields are not fully determined until their detection with the r wave and this late determination can offer explanations for non‐local entanglement.