Wigner-Eckart theorem elements

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... 

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 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]... 

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,... 

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. 

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)... 

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... 

The general matrix element over the term kets obeys the reduction (integration with respect to the angular momentum functions) according to the Wigner-Eckart theorem ... 

The Wigner-Eckart theorem states that the matrix element of a tensor operator can be expressed through a more fundamental quantity - the reduced matrix element (which is free of projections of angular momenta) and a coupling coefficient... 

As a consequence of the Wigner-Eckart theorem the replacement theorem holds true a matrix element of any irreducible tensor operator can be expressed with the help of the matrix elements formed of the angular momenta... 

Knowing the quasi-spin properties of F we can now turn our attention to the associated selection rules. According to the Wigner-Eckart theorem acting in quasi-spin space the selection of an interaction element of an operator K0 ... 

The use of quarks in atomic shell theory provides an alternative basis to the traditional one. The transformations between these bases can be complicated, but there are many special cases where our quarks can account for unusual selection rules and proportionalities between sets of matrix elements that, when calculated by traditional methods, go beyond what would be predicted from the Wigner-Eckart theorem [4,5], This is particularly true of the atomic f shell. An additional advantage is that fewer phase choices have to be made if the quarks are coupled by the standard methods of angular-momentum theory, for which the phase convention is well established. This is a strong point in favor of quark models when icosahedral systems are considered. A number of different sets of icosahedral Clebsch-Gordan (CG) coefficients have been introduced [6,7], and the implications of the different phases have to be assessed when the CG coefficients are put to use. 

Between states of well-defined quasispin Q, <2, the Wigner-Eckart theorem in quasispin space shows that any quasispin tensor has matrix elements proportional to... 

So matrix elements of U21 2 can be worked quite easy observing that U214 transform as e 2"f>. In fact using the well-known Wigner-Eckart theorem... 

As a consequence of the Wigner-Eckart theorem, relations between statistical tensors which can be derived from purely vector coupling procedures will be supplemented for transitions by introducing the corresponding reduced matrix elements Dy or Cy for the process of photoionization or Auger decay, respectively (see equs. (8.102) and (8.103b)). 

For the matrix element M dg p) entering into the righthand part of Eq. (5.8) we obtain, applying the Wigner-Eckart theorem (D.7),... 

To determine the explicit form of the matrix elements of the tensor operators we employ the Wigner-Eckart theorem [136, 379, 402]... 

The fact that the magnetic interaction Hamiltonians are compound tensor operators can be exploited to derive more specific selection rules than the one given above. Furthermore, as we shall see later, the number of matrix elements between multiplet components that actually have to be computed can be considerably reduced by use of the Wigner-Eckart theorem. 

The main reason for working with irreducible tensor operators stems from an important theorem, known as the Wigner-Eckart Theorem (WET)75,76 for matrix elements of tensor operators ... 

In systems with orbitally degenerate states, we can also exploit the Wigner-Eckart theorem for the spatial part of the wave function. Use of the WET further reduces the number of matrix elements that have to be computed explicitly. 

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