Foundations of Physics

, Volume 27, Issue 8, pp 1139–1157 | Cite as

Supermultiplicity and the relativistic Coulomb problem with arbitrary spin

  • M. Moshinsky
  • A. del Sol Mesa
  • V. Riquer
Part II. Invited Papers Dedicated to Lawrence Biedenharn


The Hamiltonian for n relativistic electrons without interaction but in a Coulomb potential is well known. If in this Hamiltonian we take r u =r′, P u =P′ with u=1,2,..., n, we obtain a one-body problem in a Coulomb field, but the appearance of n of the α u , u=1,..., n, each of which corresponds to spin\(\tfrac{1}{2}\), indicates that we may have spins up to (n/2). We analyze this last problem first by denoting the 4×4 matrices α, β as direct products of 2×2 matrices which correspond to the ordinary spin, and a new concept, also related to the SU(2) group, which we call sign spin. In this new notation our problem depends on the sixteen generators of a U(4) group reduced along the chain Û(2)⊗Ŭ(2) sub-groups associated with the ordinary and sign spins. We now make a change of variables in our Hamiltonian so a term ε related to the frequency ω of an oscillator, which will be our variational parameter, appears in it, and later construct the full states of the problem with a harmonic oscillator of frequency 1 and ordinary and sign spin parts. Finally we obtain the matrix representation of our Hamiltonian with respect to the states mentioned and discuss the energy spectra of the problem where the partition {h} representing the irrep of U(4) and j the total angular momentum, take the values {h}=[1], j=\(\tfrac{1}{2}\); {h}=[11], j=0; {h}=[2], j=0.


Harmonic Oscillator Coulomb Potential Total Angular Momentum Nonrelativistic Limit Full State 
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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • M. Moshinsky
    • 1
  • A. del Sol Mesa
    • 1
  • V. Riquer
    • 1
  1. 1.Instituto de FísicaUNAMMéxicoMéxico

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