A — Tensor Operations

we use Eq. (5.7) as an example for discussing the operations related to a tensor. The dot in Eq. (5.7) represents the multiplication of a tensor by a vector, and the product is a vector. In Eq. (5.7), the unit vector n forms a dot product with only the left unit vector in each dyad term of the tensor T. With n-x = n, n-y = Uy, and n-z = n, Eq. (5.7) is rewritten as 

Sometimes for convenience, the tensor is expressed in terms of a matrix as 

The matrix representation is convenient because dot-product multiplication is equivalent to standard matrix multiplication. In terms of the matrix representation, Eq. (5.7) is expressed as 

3) the vector n at the left of the dot is represented by a row. The multiplication of the row with the matrix forms another row, which represents a vector. In the matrix representation, the dot product between two vectors is represented by the multiplication of a row on the left and a column on the right. 

There are a few special second-order tensors, which are noteworthy (l)If 

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A tensor operator under the algebra G 3 G, T, is defined as that operator satisfying the commutation relations... 

Tx are components of a tensor operator, the explicit expression for them being... 

A tensor operator with S=0 is called a singlet operator, an operator with S... 

A comparison with Eq. (5.17) shows that at at, is a tensor operator of rank... 

The relations of Eq. (5.17) allow the scaling of all components of a tensor operator with the same factor. The factor used above is the one commonly used. It is noted that if i and j are identical the triplet operator of Eq. (5.24) vanishes. 

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... 

Having the reduced matrix element determined one can easily evaluate all the matrix elements of a tensor operator. 

Analogously for a tensor operator of rank 2 with even parity... 

In these expressions the quantity Cj,11 is proportional to the spherical harmonics ylp(f) and represents a tensor operator of rank 1 and component p. Results for both linear and circular polarizations can then be comprised in... 

Although, Jq behaves rather like a tensor operator in the molecule-fixed axis system, we must remember that it is not a tensor operator because it does not satisfy the conditions in (5.152). Elsewhere in this book, we shall often make use of this short hand. We expand a scalar product J P as... 

For a tensor operator W4 2(/fi. ki ) which acts only on, say, part 1 of a coupled scheme, the reduced matrix element is given by... 

Any quantity that transforms under rotations in the same way as the total angular momentum eigenstate jm) is called a spherical tensor T. A tensor operator of rank k is written... 

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. 

On the other hand, the modified boson operators and B are not components of a tensor operator in the sense of the above definitions. However, as shown in [31], one can use them to define a vector operator TjJ, as ... 

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... 

We have thus shown that has the properties of a tensor operator of... 

The creation operator is a tensor operator of rank I with respect to the orbital angular momentum and of rank with respect to spin. For instance. 

A tensor operator is thus created from U and V, and the analogy to the coupling of angular momentum eigenfunctions is underlined by the presence of the Clebsch-Gordan coefficient in (43) or, equivalently, the 3/ symbol in (45). If, on the other hand, we are given a double tensor, this may be represented as a sum of tensor products by... 

Theorem 12 A matrix element, involving a tensor operator, may be factorized into a product of an intrinsic scalar part and an appropriate 3F coupling coefficient. 

The result of a calculation must not depend on the coordinate system. Therefore, the result of a tensor operation must in itself be a tensor (of the appropriate order). All rules in this section fulfil this condition. A product definition of the form c = aj6j, directly multiplying the components, is not physically meaningful because the value of (ci) would depend on the coordinate system. 

Weak quadrupole perturbation of magnetic levels In this case the quadrupole interaction operator (18.51) must be projected onto the coordinate system associated with the magnetic (Zeeman) hamiltonian (18.1). Since the former is actually a tensor operator, the projection introduces a more complicated angular dependence than in the converse case considered in section 1.3.2.2. The energy levels become, on the basis of first-order perturbation theory ... 

For the relation between operators in space-fixed and molecule fixed co-ordinate systems, we use the notation that p gives the sPace fixed coordinate and q gives the moleQuIe fixed coordinate of a tensor operator. The relationship between the same tensor in the two coordinate systems is given by ... 

Step 2 The reduced matrix element of a tensor operator working only on one part... 

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