The Hurwitz-Hopf map and harmonic wave functions for integer and half–integer angular momentum

Charlista: Sergio Hojman - Departamento de Física, Facultad de Ciencias, Universidad de Chile

Resumen: Based on Schwinger’s idea of relating the raising and lowering operators of the two–dimensional harmonic oscillator to the usual angular momentum operators, I construct a different realization that recovers the well–known angular momentum operators written in terms of the three Euler angles, and produces new operators j++, j+−, j−+, and j−−, that raise and lower the total angular momentum eigenvalue j and its z component m by units of ħ/2 and another operator j which has eigenvalue j precisely. As a result, a unified description of angular momentum eigenfunctions with half–integer and integer j and m eigenvalues in terms of the three Euler angles. In addition to the results on angular momentum, I display the relation of Schwinger’s transformation to the Hopf map from R4 to R3, the Hopf fiber, the Kustaanheimo–Stiefel transformation that relates solutions to the harmonic oscillator to those of the hydrogen atom both in classical and quantum mechanics, the path–integral solution to the hydrogen atom problem, the transformation between Jones vector and Stokes parameters for polarized light, which are some of its applications in Physics and Mathematics.