Chapter 3 Representations of Groups

  • Published 2001

Abstract

The structure of abstract groups developed in Chapter 2 forms the basis for the application of group theory to physical problems. Typically in such applications, the group elements correspond to symmetry operations which are carried out on spatial coordinates. When these operations are represented as linear transformations with respect to a coordinate system, the resulting matrices, together with the usual rule for matrix multiplication, form a group that is equivalent to the group of symmetry operations in a sense to be made precise later in this chapter. In essence, these matrices form what is called a representation of the symmetry group with each element corresponding to a particular matrix. For applications to quantum mechanics, as we have seen in Section 1.2, the symmetry operations are performed on the Hamiltonian, whose invariance properties determine the symmetry group. The wavefunctions, which do not all share the symmetry of the Hamiltonian, will be seen to determine the representations of the symmetry group in the sense described above. These representations will, in turn, provide a classification scheme for the eigenfunctions of the Hamiltonian, in

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@inproceedings{2001Chapter3R, title={Chapter 3 Representations of Groups}, author={}, year={2001} }