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Keywords:

  • symmetry;
  • equivalences;
  • eigenvalue equations;
  • stationary equations;
  • spatial localization

Abstract

A generalization of the concept of Wigner's symmetrical operator, i.e., of the operator invariant under symmetry transformations forming a group, namely equation image, is proposed here. Symmetrical operators in the present general sense do constitute a set whose elements are mutually commuting and equivalent to each other under transformations equation image. A generalization of a well known theorem attributed to Wigner is established: for a given operator  and its equivalent ones Â[i] = Ŝmath image ÂŜi, symmetrical as previously defined, it is always possible to find a complete set of common eigenfunctions that form canonical basis sets for representations of equation image induced by the largest subgroup equation image whose elements keep  invariant (the particular case equation image reduces to Wigner's theorem itself). The formal generalization presented here opens the possibility of describing and exploiting two complementary aspects of symmetry: invariance and equivalence, within a unified theoretical approach. It is also shown that the symmetry properties of the stationary solutions of Boys' localization functional obey a straightforward extension of this theorem. A few examples are provided to highlight the implications of this alternative way to look at symmetry in quantum systems. © 2012 Wiley Periodicals, Inc.