Standing Waves

This is an discussion of Schroedinger's 1926 paper. The subsequent notions of 'spin groups' and 'gauge cocycles' arose because O(1) is not equivariant for automorphims of the Riemann sphere of angles; understanding this requires reconciling the Legendre and Weyl operators, the latter which are infinitesimal, operating on functions of angles. Notions of 'spin up' and 'spin down' are different, referring to character values. Rather than tensoring with an abstract 'spin' representation of SU_2 it makes more sense to understand the classical Schroedinger solutions as the images under contraction against the Euler derivation of differential two-forms, then restricted to the Severi-Brauer variety of formal angles. The various orbit-orbit coupling schemes can be unified into a single scheme based on a tree with branching at eight levels; as for spin-orbit coupling, this is explained by the notion that the Laplace operator should have been acting on diagonal coordinates. The group SU_2xSU_2 arises naturally here as a semidirect product of automorphisms of base and fiber of a vector bundle, and the corresponding change in the Schroedinger equation gives a different calculation of the fine structure. A javascript at http://spectrograph.uk uses a character calculation to obtain the perturbation approximation of the spectrum of every atom and ion, and shows things in the visible spectrum like the Northern Lights, the green Magnesium star triplet, and the Sodium doublet. It is shown that Schroedinger's analogy with the Hamiltonian approach would require further relativistic work, and the meaning of such an enterprise is given ethical considerations also.