Tensor Operators and the Wigner-Eckart Theorem

Abstract

AbstractIn this chapter we pave the way to the use of the coupling methods of Chap. 1 for manipulating operators and their matrix elements. To enable smooth application of the angular momentum methods, we introduce so-called spherical tensor operators. Spherical tensors can be related to Cartesian tensors. A Cartesian tensor of a given Cartesian rank can be reduced to spherical tensors of several spherical ranks. There is a very convenient procedure, the so-called Wigner-Eckart theorem, to separate the part containing the projection quantum numbers from the rest of the matrix element of a spherical tensor operator. The remaining piece, called the reduced matrix element, is rotationally invariant and contains the physics of the matrix element.