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

Sergio A. Hojman, Eduardo Nahmad-Achar, Adolfo Sánchez-Valenzuela

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Article number085249
JournalPhysica Scripta
Issue number8
StatePublished - 1 Aug 2023
Externally publishedYes


  • Hurwitz-Hopf map
  • harmonic wave functions
  • integer and half-integer angular momentum
  • supersymmetry


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