The Hurwitz-Hopf Map and Harmonic Wave Functions for Integer and
Half-Integer Angular Momentum
- URL: http://arxiv.org/abs/2211.10775v2
- Date: Thu, 8 Jun 2023 01:07:21 GMT
- Title: The Hurwitz-Hopf Map and Harmonic Wave Functions for Integer and
Half-Integer Angular Momentum
- Authors: Sergio A. Hojman, Eduardo Nahmad-Achar, and Adolfo
S\'anchez-Valenzuela
- Abstract summary: Harmonic wave functions for integer and half-integer angular momentum are given in terms of the angles $(theta,phi,psi)$ that define a rotation in $SO(3)$.
A new nonrelistic quantum (Schr"odinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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