Applications

Abstract

Standard framework of the symmetry application in physics has three main parts. First, the description of the system is formalized within the state spaceState space ; it is a vector space, with the vectors representing states of the system. Actually, a physical problem under consideration determines the state space . For instance, in quantum mechanical studies, S is a Hilbert space of the wave functions of the system, while in the vibrational analyses it is the space of the all possible atomic displacements. The second part is symmetry group L and its representation D(L) in the state space; i.e., the set of the operators D(ℓ) in S corresponding to the group elements. Finally, the main task is usually to solve the eigenproblem of the hamiltonian H, a hermitian operator in the state space which governs the dynamics. Choice of H depends on the specific problem considered, as well as on the approximations involved in the physical model. Hamiltonian and symmetry commute: D(ℓ)H = HD(ℓ) for each symmetry transformation ℓ. © 2010 Springer-Verlag Berlin Heidelberg.