Theorem Wigner-Eckart

After arranging our notation to accommodate spherical tensors we turn to a powerful tool for the calculation of matrix elements. The Wigner-Eckart theorem allows the factorisation of a matrix element into a part containing the dependence of the magnetic quantum numbers, essentially a 3j symbol, and a part independent of these, called the reduced matrix element and already employed in sect. 2  

In order to achieve a more compact notation we make the abbreviations 6 = v,S,L, 6 = v, S, L etc. as in sect. 2. Again, with out main interest centered on the lanthanide series, we have used in (52) eigenstates of the total angular momentum operators J, E in the matrix element. 

The coupling of Russell-Saunders states 6, M, Mj ) by means of Clebsch-Gordan coefficients, 

We note for later applications of this formula that, k in (54) will be restricted to the values 1 and 0. Also, the operators u operate in different spaces, e.g., the orbital or the spin part of a wavefunction. Note, furthermore, that the reduced matrix element on the right-hand side of (54) contains only a product of tensor components and not a tensor product. 

If a reduced matrix element between many-electron states is to be expressed in terms of those of single-electron states, we have to introduce the concept of fractional parentage coefficients pioneered by Racah. Collecting the general parts of a reduced matrix element between states with n equivalent electrons, we are led to the definition of a Racah tensor and, in this case, the fractional parentage coefficients will appear explicitly only in this definition  

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In the latter expression the matrix element of operator dq> is transformed according to the Wigner-Eckart theorem and the definition used is... 

The matrix element of operator is written in terms of the Wigner-Eckart theorem, and the integral part is denoted as... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

Another important property of the angular momentum is the Wigner-Eckart theorem.2 This theorem states that the matrix elements of any tensor operator can be separated into two parts, one containing the m dependence and one independent of m,... 

The angular part depends only on properties of the angular momentum. Using the Wigner-Eckart theorem, one has... 

At this point, we should invoke the so-called Wigner-Eckart theorem, whose demonstration is beyond the scope of this book (see, e.g., Tsukerblat, 1994). From this theorem, it is possible to establish the following selection rule ... 

The last two matrix elements in Eq. (8.45) can be evaluated using the Wigner-Eckart theorem [5] ... 

From the Wigner-Eckart theorem, these matrix elements may be written in terms of 3-j symbols, as [350]... 

The transition operator J t) is determined by the perturbed Hamiltonian H and, in particular, transforms in the same manner as T does under the symmetry group of H0. The manner in which J(t) and Y transform determines selection rules, through the use of the Wigner-Eckart theorem. 

Selection rules arise on considering how the transition operator f(t) transforms under the double group SF El We note that if f(t) is spin free it transforms the same way under SF El ° a as it does under S SF. Selection rules follow from the Wigner-Eckart theorem, just as for the spin-free case discussed in Section III. Selection rules for operators which contain spin may also be derived on considering22 how such operators transform in S SF El... 

A general development of matrix elements of an irreducible tensor operator leads to the Wigner-Eckart theorem (see, for example, Tinkham [2] or Chisholm [7]), which relates matrix elements between specific symmetry species to a single reduced matrix element that depends only on the irrep labels, but this is beyond the scope of the present course. 

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,... 

As was mentioned in the previous paragraph, the Wigner-Eckart theorem (5.15) is fairly general, it is equally applicable for both approaches considered, for tensorial operators, acting in various spaces (see, for example, Chapters 15,17 and 18, concerning quasispin and isospin in the theory of an atom). 

Usually the Wigner-Eckart theorem (5.15) is utilized to find the dependence of the matrix elements on the projections of angular and spin momenta. Its use in the quasispin space... 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

Utilization of the tensorial properties of the electron creation and annihilation operators allows us to obtain expansions in terms of irreducible tensors of any operators in the second-quantization representation. So, using the Wigner-Eckart theorem (5.15) in (14.11) and (14.12), then coupling ranks of second-quantization operators by (5.12) and utilizing (14.10), we can represent one-shell operators of angular momentum in the irreducible tensor form... 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

The matrix element is defined relative to two-electron wave functions of coupled momenta. If now we take into account the tensorial structure of operator (14.57), apply the Wigner-Eckart theorem to this matrix element and sum up over the appropriate projections, we have... 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

Applying the Wigner-Eckart theorem in all three spaces, we establish the property of SCFP in relation to interchanges of the spin and quasispin quantum numbers... 

These tensorial products have certain ranks with respect to the total orbital, spin and quasispin angular momenta of both shells. We shall now proceed to compute the matrix elements of such tensors relative to multi-configuration wave functions defined according to (17.56). Applying the Wigner-Eckart theorem in quasispin space to the submatrix element of tensor gives... 

For the basis (18.27) to be used effectively in practical computations an adequate mathematical tool is required that would permit full account to be taken of the tensorial properties of wave functions and operators in their spaces. In particular, matrix elements can now be defined using the Wigner-Eckart theorem (5.15) in all three spaces, so that the submatrix element will be given by... 

The energy operators are scalar, therefore, their matrix elements, according to the Wigner-Eckart theorem (5.15), are diagonal with respect to the total momenta and do not depend on their projections. For these reasons we shall skip projections further on. The expression for the matrix elements of the sum of operators (19.5) and P in (1.15) is simply equal to its one-electron matrix element, multiplied by N, i.e. 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

If we remember that isospin behaves as angular momentum in a certain additional space, we shall be able to apply the Wigner-Eckart theorem to the matrix elements of appropriate tensors in that space, and also the entire technique of the SU2 group. So, having applied this theorem to a certain matrix element in both spaces, we obtain... 

Equation (24.3) is also valid for intermediate coupling. Applying the Wigner-Eckart theorem (5.15) to (24.1) and performing the summation, we arrive at the result... 

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