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 in ${\mathbb R}^3$. 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. 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, and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of $G_0$. 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 may be solved exactly in terms of the harmonic wave functions hereby introduced.