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Appendices

  • Ulrich Rössler
Part of the Advanced Texts in Physics book series (ADTP)

Abstract

Geometrical operations (translation, rotation, inversion) which leave a geometrical object (here the crystal lattice) invariant, are symmetry operations. Mathematically they form a group, the symmetry group of the crystal: for the translations it is the translation group, for the rotations, inversion, and their combinations it is the point group. The number g of elements in the group is its order. The symmetry of a system implies the invariance of the system hamiltonian H (be it for phonons, electrons, or magnons) under unitary operations corresponding to the geometrical operations of the symmetry group. These unitary operations form a group which is isomorphic to the symmetry group. When applied to a set of eigenfunctions of H this set is transformed into another set of eigenfunctions, which can be represented as a linear combination of the former ones. The eigenfunctions of a degenerate eigenvalue transform among each other and form an invariant subspace in the Hilbert space of H. In a chosen basis, these unitary operations can be formulated as matrices which define another group isomorphic to the symmetry group. For a proper choice of the basis, all matrices of the matrix representation have block-diagonal form with the dimension of the block matrices indicating the degeneracy of the invariant subspaces.

Keywords

Symmetry Group Irreducible Representation Invariant Subspace Point Group Symmetry Operation 
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

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Ulrich Rössler
    • 1
  1. 1.Institut für Theoretische PhysikUniversität RegensburgRegensburgGermany

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