Reduced Wigner matrices

Thus, the director motion is solely described by the Euler angle / ". When small angle director fluctuations are assumed (i.e., / " is small such that sin0 = ), the only non-zero reduced Wigner matrices (see Appendix A) are... 

The matrix element of e is the scalar quantity which is termed the reduced Wigner matrix. 

Table 1. The values of the reduced Wigner matrix elements dmn(l3 Definition D (a, p, y) =...

Table 1. Second-rank reduced Wigner matrix elements...

The reduced density-matrix element is defined in terms of T by the Wigner—Eckart theorem (3.104). 

Fig. 19.1 provides a concise summary of these relationships. A more elaborate figure that adds the connections to the Wigner [38,39] and Moyal [40] mixed position-momentum representations of the first-order reduced density matrix can be found in an article that also works out all these functions in closed form for a simple harmonic model of the helium atom [41]. 

The subscript W refers to this partial Wigner transform, N is the eoordinate space dimension of the bath and X = R, P). In this partial Wigner representation, the Hamiltonian of the system takes the form Hw R,P) = P /2M + y-/2m+ V q,R). If the subsystem DOF are represented using the states of an adiabatic basis, a P), which are the solutions of hw R) I R)=Ea R) I where hw K)=p /2m+ V q,R) is the Hamiltonian for the subsystem with fixed eoordinates R of the bath, the density matrix elements are p i -, 0 = ( I Pw( 01 )- From the solution of the quantum Liouville equation given some initial state of the entire quantum system, the reduced density matrix elements of the quantum subsystem of interest can be obtained by integrating over the bath variables, p f t) = dX p X,t), in order to find the populations and off-diagonal elements (coherences) of the density matrix. 

RE field along an axis of the rotating frame C = spin-rotation interaction tensor (Q) = reduced Wigner rotation matrix elements ... 

In this field, the density operator plays an important and uncontested role. It allows for more than just the above-given series expansion it can be used for consistent approximations by integrating out (read take the partial trace) of a series of states we are not particularly interested in, leading to the so-called reduced density matrix. It can also be used to find representations in spaces, for instance, the Wigner representation [35], that give more insight into the quantum distribution functions, and provide in some cases distribution functions that are more close to the classical. 

The reduced matrix elements of the spherical harmonics can be written in terms of a Wigner 3 - j symbol... 

By Wigner-Eckart s theorem [6] Eq. (2) can be expressed in terms of a reduced matrix element that is independent of M and 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. 

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, 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)). 

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. 

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. 

Only one of the matrix elements needs to be evaluated explicitly. All others can be obtained from the reduced matrix element by means of the Wigner-Eckart theorem, Eq. [166]. The eigenvectors... 

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. 

This is the Wigner-Eckart theorem, a very important result which underpins most applications of angular momentum theory to quantum mechanics. It states that the required matrix element can be written as the product of a 3- j symbol and a phase factor, which expresses all the angular dependence, and the reduced matrix element (rj, j T/ (d) if. j ) which is independent of component quantum numbers and hence of orientation. Thus one quantity is sufficient to determine all (2j + 1) x (2k + 1) x (2/ + 1) possible matrix elements (rj, j, mfIkq(A) rj, jf m ). The phase factor arises because the bra (rj, j, m transforms in the same way as the ket (— y m rj, j, —m). The definition of the reduced matrix element in equation (5.123), which is due to Edmonds [1] and also favoured by Zare [4], is the one we shall use throughout this book. The alternative definition, promoted by Brink and Satchler [3],... 

We next consider how to evaluate the reduced matrix element. Although it is defined in equation (5.123), this is not a useful relationship for its evaluation. The usual approach is to calculate the matrix element and then derive rj, j T (A) rj, /) from the result. The evaluation of the reduced matrix element for the total angular momentum J itself is a good example. From the Wigner-Eckart theorem, we write... 

Applying the Wigner-Eckart theorem to this equation, we obtain an expression for the reduced matrix element ... 

Reduced matrix elements are defined by using the Wigner-Eckart theorem to evaluate the dependence on projection quantum numbers ... 

For the purely nuclear term we have to evaluate the reduced matrix element of T2( Q) in (8.491), and to achieve this we make use of the Wigner-Eckart theorem ... 

The reduced matrix element in (9.12) is evaluated by noting the definition of the nuclear quadrupole moment gN, and using the Wigner Eckart theorem as follows ... 

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