Tensors Wigner-Eckart theorem

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,... 

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. 

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... 

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... 

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 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... 

In this section we first present a set of general transformation formulae for tensor operators associated with SRMs. These then serve as a mathematical tool for the formulation of Wigner-Eckart theorems and selection rules for irreducible tensor operators associated with multipole transitions of SRMs. The concept of isometric groups will allow a formulation of selection rules in strict analogy to the group theoretical treatment of quasirigid molecules first presented by Wigner5. ... 

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... 

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)). 

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 ... 

There is a first-order splitting pattern common to all 3n states, independent of the physical content. All the molecule-dependent physical information is contained in the parameter Aso- These facts are, of course a consequence of the tensor properties expressed in the Wigner-Eckart theorem. 

The desired coupled basis will be performed by the methods given by Racah and Wigner. Making use of the Wigner-Racah formalism and of the Wigner-Eckart theorem and observing some rules for the matrix elements of the products of tensor operators, we obtain for the matrix elements of the quadrupole interaction operator/Z ... 

Da nunmehr eine Komponente des Quadrupolmoment-Tensors bekannt ist, kann das Wigner-Eckart-Theorem angewandt werden. Man erhalt so das Redu-zierte Matrixelement... 

Matrix elements of spherical tensor operators the Wigner-Eckart theorem... 

We can also form a second rank tensor by coupling the angular momentum J with itself. In this case, the Wigner Eckart theorem is expressed as ... 

Another immediate corollary of the Wigner-Eckart theorem is the replacement theorem, which allows one to write the matrix elements of one spherical tensor operator,... 

The Wigner-Eckart theorem (Biedenharn and Louck, 1981a, p. 96) can be used to obtain Eq. (40) directly. According to this theorem the matrix elements in an angular momentum basis of a spherical tensor operator V (q = —k, — k + 1,k) have a particularly simple structure given by3... 

The matrix elements of the tensor operator Uq which are diagonal in the spin S can be rewritten using the Wigner-Eckart theorem with the 3-j symbols and the reduced matrix elements... 

The dependence on the m indices of the amplitude for the transition between states JM) and J M ) of total angular momentum due to a tensor operator Tq has a remarkably simple form in which the indices M, M and Q all appear in a single 3-j symbol. It is given by the Wigner—Eckart theorem. 

The simplest example of the Wigner—Eckart theorem is given by the Gaunt integral over three spherical harmonics, which is the matrix element for the transition between eigenstates m) and fm ) of a single orbital angular momentum observable due to a tensor operator Tj. We prefer to use the renormalised tensor operator C, which simplifies the expression. 

The physical meaning of the first-rank tensor can be seen when these are related to the components of the mean angular momentum vector (/). With the help of the Wigner—Eckart theorem one can show that... 

Consider now the selection rules for the minimal value l for which y and a are nonzero. In order to do this, it is convenient to pass from the spherical components of the tensors Q/ni and ajn2 to the components of irreducible tensor operators <2r,y, and aV]yj that transform after the irreducible representation contained in the decomposition of D and DJ, respectively. If /iry) are the different vibronic states transforming after the representation T and corresponding to the vibronic level with the energy e r, then for the collision-induced spectrum using the Wigner-Eckart theorem, we have... 

One of our main motivations for pursuing the development of a density functional response theory for open-shell systems has been to calculate spln-Hamiltonian parameters which are fundamental to experimental magnetic resonance spectroscopy. It is only within the context of a state with well-defined spin we can speak of effective spin Hamiltonians. The relationship between microscopic and effective Hamiltonians rely on the Wigner-Eckart theorem for tensor operators of a specific rank and states which transform according to their irreducible representations [45]. 

Since many of the operators that appear in the exact Hamiltonian or in the effective Hamiltonian involve products of angular momenta, some elementary angular momentum properties are summarized in the next section. Matrix elements of angular momentum products are frequently difficult to calculate. A tremendous simplification is obtained by working with spherical tensor operator components and, in this way, making use of the Wigner-Eckart Theorem (Section 3.4.5). A more elementary but cumbersome treatment, based on Cartesian operator components, is presented in Section 2.3. 

The power of the Wigner-Eckart theorem (Messiah, 1960, p. 489 Edmonds, 1974, p. 75) is that it relates one nonzero matrix element to another, thereby vastly reducing the number of integrals that must either be explicitly evaluated or treated as a variable parameter in a least-squares fit to spectral data. For example, consider S k a tensor operator of rank k that acts exclusively on spin variables. The Wigner-Eckart theorem requires... 

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