Transformation Properties of the Wave Function

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

Alternatively, we can view the angles 0 and ]) as defining the orientation of a general rotation axis z , about which a rotation through v / is to be performed, relative to a space-fixed coordinate system x, y, z. 

The first and the third of the Euler angles, j) and /, refer to rotations about z axes. The corresponding transformation matrices in 3D coordinate space read 

A rotation through the Euler angle 0 refers to the y axis and is represented by 

These three rotations do in general not commute, that is, the product depends on the order in which the matrices are multiplied. The representation matrix of the complete transformation can be composed from these individual rotation matrices as 

---

The expressions for eigenvalues of the Casimir operators are presented below. They are completely defined by the transformation properties of the wave functions with regard to the corresponding groups and depend only on indices characterizing the representations of these groups. For the cases discussed above they are, respectively... 

---