Foundations of Physics

, Volume 27, Issue 8, pp 1085–1104 | Cite as

Doubly stochastic matrices in quantum mechanics

  • James D. Louck
Part II. Invited Papers Dedicated to Lawrence Biedenharn


The general set of doubly stochastic matrices of order n corresponding to ordinary nonrelativistic quantum mechanical transition probability matrices is given. Landé's discussion of the nonquantal origin of such matrices is noted. Several concrete examples are presented for elementary and composite angular momentum systems with the focus on the unitary symmetry associated with such systems in the spirit of the recent work of Bohr and Ulfbeck. Birkhoff's theorem on doubly stochastic matrices of order n is reformulated in a geometrical language suitable for application to the subset of quantum mechanical doubly stochastic matrices. Specifically, it is shown that the set of points on the unit sphere in cartesian n!-space is surjective with the set of doubly stochastic matrices of order n. The question is raised, but not answered, as to what is the subset of points of this unit sphere that correspond to the quantum mechanical transition probability matrices, and what is the symmetry group of this subset of matrices.


Angular Momentum Total Angular Momentum Transition Probability Matrix Spin Projection Hermitian Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • James D. Louck
    • 1
  1. 1.Los Alamos National LaboratoryTheoretical DivisionLos Alamos

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