Wigner projection operators

The use of redundant coordinates requires extensive modification of the lattice dynamical procedure. It is, however, often worth the additional complication to use redundancies if this facilitates the formulation of symmetry coordinates. When the Wigner projection operator (Wigner, 1931) is used to build such symmetry coordinates, it is necessary to first understand the results of the application of all symmetry operations of the applicable group to the displacement coordinates chosen. This is indeed relatively straightforward for the direction cosine displacement coordinates and therein lies their principal value. These coordinates transform like axial vectors in contrast to cartesian coordinates, which transform like polar vectors. 

QTST is predicated on this approach. The exact expression 50 is seen to be a quantum mechanical trace of a product of two operators. It is well known, that such a trace can be recast exactly as a phase space integration of the product of the Wigner representations of the two operators. The Wigner phase space representation of the projection operator limt-joo %) for the parabolic barrier potential is h(p + mwtq). Computing the Wigner phase space representation of the symmetrized thermal flux operator involves only imaginary time matrix elements. As shown by Poliak and Liao, the QTST expression for the rate is then ... 

The spin term is straightforward to evaluate by the Wigner-Eckart theorem but the rotational term requires further consideration. Let us introduce the projection operator onto the complete set of rotational functions between the operators Pp (J) and 2) (< >) (the closure relationship) ... 

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,... 

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... 

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. 

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... 

Wigner, as research director at Clinton, assumed responsibility for the project. In this he was, unfortunately, not supported very well by the old timers (meaning those of us who had already been there for a couple of years) we could not understand how uranium at 100 per gram (1946 ) could ever be an economical fuel and besides, we were too involved with the MTR. Perhaps this lack of support was an important reason why the Daniels pile was never built, although many similar reactors have since been operated successfully. 

The construction of spin-free CFs that transform, under electron permutations, like the basis vectors of an irrep of S, has been considered already in Section 7.6, and it remains only to fill in the details. Such functions are generated by projection , using Wigner operators, and, because any permutation of variables in an orbital product is equivalent to the inverse permutation of orbital indices, it is possible to classify the CFs by means of Weyl tableaux, which contain indices (selected from 1,2,..., m) instead of electron labels (1,2,..., N). The functions to be used may thus be constructed, from the product Q = k k2... k, as follows ... 

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