This article is dedicated in memory of Prof. Cesare Pisani.
Beyond Wigner's theorems: The role of symmetry equivalences in quantum systems†
Article first published online: 5 JUN 2012
Copyright © 2012 Wiley Periodicals, Inc.
International Journal of Quantum Chemistry
Special Issue: Mexican Theoretical Physical Chemistry Meetings
Volume 112, Issue 21, pages 3543–3551, 5 November 2012
How to Cite
Zicovich-Wilson, C. M. and Erba, A. (2012), Beyond Wigner's theorems: The role of symmetry equivalences in quantum systems. Int. J. Quantum Chem., 112: 3543–3551. doi: 10.1002/qua.24184
- Issue published online: 18 SEP 2012
- Article first published online: 5 JUN 2012
- Manuscript Accepted: 18 APR 2012
- Manuscript Revised: 5 APR 2012
- Manuscript Received: 21 FEB 2012
- eigenvalue equations;
- stationary equations;
- spatial localization
A generalization of the concept of Wigner's symmetrical operator, i.e., of the operator invariant under symmetry transformations forming a group, namely , 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 . A generalization of a well known theorem attributed to Wigner is established: for a given operator Â and its equivalent ones Â[i] = Ŝ 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 induced by the largest subgroup whose elements keep Â invariant (the particular case 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.