Tensorial products and their matrix elements

Let us present the main definitions of tensorial products and their matrix or reduced matrix (submatrix) elements, necessary to find the expressions for matrix elements of the operators, corresponding to physical quantities. The tensorial product of two irreducible tensors and is defined as follows  

A fundamental role is played in theoretical atomic spectroscopy by the Wigner-Eckart theorem, the utilization of which allows one to find the dependence of any matrix element of an arbitrary irreducible tensorial operator on projection parameters, 

The quantity denoted ( ) is called a reduced matrix (submatrix) element of operator T k). It does not depend on projection parameters m, m, q. Dependence of the matrix element considered on these projections is contained in one Clebsch-Gordan coefficient. Such dependence is one of the indicators of the exceptional role played by the Clebsch-Gordan coefficients in the theory of many-particle systems. Their definitions and main properties will be discussed in the next paragraph. 

The submatrix element of the tensorial product of two operators, acting on one and the same coordinate, may be calculated applying the formula 

The last multiplier on the right side of (5.19) is the 97-coefficient, defined in Chapter 6. Such expressions for submatrix elements directly follow from (5.19) in the case when the operator acts only on coordinates with index 1 or 2  

---

Operators of electronic transitions, except the third form of the Ek-radiation operator for k > 1, may be represented as the sums of the appropriate one-electron quantities (see (13.20)). Their matrix elements for complex electronic configurations consist of the sums of products of the CFP, 3n./-coefficients and one-electron submatrix elements. The many-electron part of the matrix element depends only on tensorial properties of the transition operator, whereas all pecularities of the particular operator are contained in its one-electron submatrix element. 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

It is clear then that one needs to describe all relevant operators (related to the spin components) using their matrix representations in the four-fold vector space generated by the state vectors mj, ms). These matrices can be constructed by evaluating each matrix element or they can be built by the direct tensorial product of the corresponding 2x2 matrices that describe the dynamics of the separate spin 1 /2 systems [5,12], The direct tensorial product of two nxn matrices A and B lead to a x matrix C = A B whose elements are... 

---