| Abstrakt: |
Our second general application of group theory to physical problems will be to selection rules. In considering selection rules we always involve some interaction Hamiltonian matrix H' that couples two states ψα and ψβ. Group theory is often invoked to decide whether or not these states are indeed coupled and this is done by testing whether or not the matrix element (ψα,H'ψβ) vanishes by symmetry. The simplest case to consider is the one where the perturbation H' does not destroy the symmetry operations and is invariant under all the symmetry operations of the group of the Schrödinger equation. Since these matrix elements transform as scalars (numbers), then (ψα,H'ψβ) must exhibit the full group symmetry, and must therefore transform as the fully symmetric representation Γ1. Thus, if (ψα,H'ψβ) does not transform as a number, it vanishes. To exploit these symmetry properties, we thus choose the wave functions ψα and ψβ to be eigenfunctions for the unperturbed Hamiltonian, which are basis functions for irreducible representations of the group of Schrödinger's equation. Here H'ψβ transforms according to an irreducible representation of the group of Schrödinger's equation. This product involves the direct product of two representations and the theory behind the direct product of two representations will be given in this chapter. If H'ψβ is orthogonal to ψα, then the matrix element (ψα,H'ψβ) vanishes by symmetry; otherwise the matrix element need not vanish, and a transition between state ψα and ψβ may occur. [ABSTRACT FROM AUTHOR] |
