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

Abstract

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 ${\mathbb R}^3$,
keeping the usual meaning of the spherical coordinates $(r,\theta,\phi)$.
They form a Hilbert (super)-space decomposed in the form 
$\mathcal H=\mathcal H_0\oplus\mathcal H_1$.
Following a classical work 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 $\mathcal H$ 
is approached from the double covering group homomorphism $SU(2)\to SO(3)$
and the topology involved is taken care of by using 
the Hurwitz-Hopf map $H:{\mathbb R}^4\to{\mathbb R}^3$. 
It is shown how to reconsider $H$ as a 2-to-1 group map, 
$G_0={\mathbb R}^+\times SU(2)\to {\mathbb R}^+\times SO(3)$,
translating it into an assignment $(z_1,z_2)\mapsto (r,\theta,\phi,\psi)$ 
whose domain consists of pairs $(z_1,z_2)$ of complex variables,
under the appropriate identification of ${\mathbb R}^4$ with ${\mathbb C}^2$.
It is shown how the Lie algebra of $G_0$ is coupled with two Heisenberg
Lie algebras of $2$-dimensional (Schwinger's) harmonic oscillators
generated by the operators $\{z_1,z_2,\bar{z}_1,\bar{z}_2\}$ 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 $\mathcal H$ can be written in terms of
polynomials in the complex coordinates $(z_1,z_2)$
and their complex conjugates $(\bar{z}_1,\bar{z}_2)$
and the representations are explicitly constructed via 
the various highest weight (or lowest weight)
vector representations of $G_0$.
Finally, a new non–relativistic quantum (Schr"odinger-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.