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

Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (theta, phi,psi) that define a rotation in SO(3), and the Euclidean norm r in R-3, keeping the usual meaning of the spherical coordinates (r,theta, phi). They form a Hilbert (super)-space decomposed in the form Eta = Eta(0) circle plus Eta(1). Following a classicalwork by Schwinger, 2-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space. is approached from the double covering group homomorphism SU(2)-> SO(3) and the topology involved is taken care of by using theHurwitz-Hopfmap H:. R-4 -> R-3. It is shown howto reconsiderHas a 2-to-1 groupmap, G0 =R+ Chi SU (2)-> R(+)x SO(3), translating it into an assignment (z(1), z(2)) |->(r,theta, phi,psi) whose domain consists of pairs (z1, z2) of complex variables, under the appropriate identification of. 4 with.2. It is shown howthe Lie algebra ofG0 is coupled with two Heisenberg Lie algebras of 2-dimensional (Schwinger's) harmonic oscillators generated by the operators {z1, z2, (z) over bar (1), (z) over bar (2)} and their adjoints. The whole set of operators gets algebraically closed either into a 13-dimensional Lie algebra or into a (4|8)-dimensional Lie superalgebra. The wave functions in. can be written in terms of polynomials in the complex coordinates (z(1), z(2)) and their complex conjugates ((z) over bar (1), (z) over bar (2)) and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations ofG(0). Finally, a new nonrelativistic quantum (Schrodinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of (r,theta, phi,psi) and the time t. The equation is succeptible to be solved exactly in terms of the harmonic wave functions hereby introduced.