The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum
Year: 2023
Type: article
Abstract: Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles (θ, ϕ, ψ) that define a rotation in SO(3), and the Euclidean norm r in <?CDATA ${{\mathbb{R}}}^{3}$?> R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form <?CDATA ${ \mathcal H }={{ \mathcal H }}_{0}\oplus {{ \mathcal H }}_{1}$?> = 0 ⊕ 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 <?CDATA ${ \mathcal H }$?> is approached from the double covering group homomorphism SU(2) → SO(3) and the topology involved is taken care of by using the Hurwitz-Hopf map <?CDATA $H:{{\mathbb{R}}}^{4}\to {{\mathbb{R}}}^{3}$?> H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, <?CDATA ${G}_{0}={{\mathbb{R}}}^{+}\times {SU}(2)\to {{\mathbb{R}}}^{+}\times {SO}(3)$?> G 0 = R + × SU ( 2 ) → R + × SO ( 3 ) , translating it into an assignment (z 1, z 2) ↦ (r, θ, ϕ, ψ) whose domain consists of pairs (z 1, z 2) of complex variables, under the appropriate identification of <?CDATA ${{\mathbb{R}}}^{4}$?> R 4 with <?CDATA ${{\mathbb{C}}}^{2}$?> 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 <?CDATA $\{{z}_{1},{z}_{2},{\bar{z}}_{1},{\bar{z}}_{2}\}$?> { z 1
Source: Physica Scripta
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,220
OpenAlex ID: W4381614079