ABSTRACT
In the language, and within the understanding, of the quantum mechanics the symmetry of a many particle system is determined by the transformation properties of the Hamiltonian. Among the collection of all transformations those that commute with the Hamiltonian are symmetry operations. The commutation relations mean that the Hamiltonian remains invariant under the action of the symmetry operations. Keeping in mind our future applications of the tools of quantum mechanics for evaluation of various matrix elements, in the case of free atoms and ions, the most important are the symmetry operations of the rotation group in three dimensions, SO(3). The angular momentum operators are the generators of all the rotations in three dimensional space. As a consequence, the Hamiltonian of spherically symmetric systems have to commute with the generators of the symmetry operators. In this particular case it means that Hamiltonian has to commute with the angular momentum operators. Consequently, the basic property of the hermitan operators states that if two operators commute, they have a common set of the eigenfunctions (but not the same eigenfunctions!) and simultaneously measured eigenvalues. As a result the energy states carry also information about the eigenvalues of angular momentum operators (only of these that commute with the Hamiltonian). These transformation properties, distinguished by the eigenvalues of appropriate angular momentum operators, are the formal source of the quantum numbers that identify the energy states of a many electron system. At the same time, the spherical harmonics form the basis sets for the irreducible representations of the three dimensional rotation group. In the case of the systems of symmetry lower than spherical, as in the case of crystals, the transformation properties are defined by the point groups, and the identification of the energy states is also based on the commutation relations of the Hamiltonian and the symmetry operators. In such cases, instead of quantum numbers, the irreducible representations of a given group directly identify the energy levels.
