TY - JOUR

T1 - The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum

AU - Hojman, Sergio A.

AU - Nahmad-Achar, Eduardo

AU - Sánchez-Valenzuela, Adolfo

N1 - Funding Information:
It is a pleasure to thank Octavio Castaños–Garza and Luis F Urrutia for fruitful conversations in the early stages of this work. Two of the authors acknowledge supports received by their research grants. EN-A acknowledges partial support by DGAPA-UNAM under project IN100323. OA-SV acknowledges CONACYT Grant #A1-S-45886.
Publisher Copyright:
© 2023 The Author(s). Published by IOP Publishing Ltd.

PY - 2023/8/1

Y1 - 2023/8/1

N2 - 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 R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form = 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 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 H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, 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 R 4 with 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 , z ¯ 1 , 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 can be written in terms of polynomials in the complex coordinates (z 1, z 2) and their complex conjugates ( z ¯ 1 , z ¯ 2 ) and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of G 0. Finally, a new nonrelativistic quantum (Schrödinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of (r, θ, ϕ, ψ) and the time t. The equation is succeptible to be solved exactly in terms of the harmonic wave functions hereby introduced.

AB - 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 R 3 , keeping the usual meaning of the spherical coordinates (r, θ, ϕ). They form a Hilbert (super)-space decomposed in the form = 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 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 H : R 4 → R 3 . It is shown how to reconsider H as a 2-to-1 group map, 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 R 4 with 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 , z ¯ 1 , 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 can be written in terms of polynomials in the complex coordinates (z 1, z 2) and their complex conjugates ( z ¯ 1 , z ¯ 2 ) and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of G 0. Finally, a new nonrelativistic quantum (Schrödinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of (r, θ, ϕ, ψ) and the time t. The equation is succeptible to be solved exactly in terms of the harmonic wave functions hereby introduced.

KW - Hurwitz-Hopf map

KW - harmonic wave functions

KW - integer and half-integer angular momentum

KW - supersymmetry

UR - http://www.scopus.com/inward/record.url?scp=85167462392&partnerID=8YFLogxK

U2 - 10.1088/1402-4896/ace08e

DO - 10.1088/1402-4896/ace08e

M3 - Article

AN - SCOPUS:85167462392

SN - 0031-8949

VL - 98

JO - Physica Scripta

JF - Physica Scripta

IS - 8

M1 - 085249

ER -