Wigner operator

The relevance of quasi-distributions in physical applications has been discussed by Dahl [23]who has shown that the Wigner function is the only one which satisfies strong requirements sueh as being the expectation value of the so-called Wigner operator and therefore to be a description based on observables in the sense of Dirae [24]. For one partiele, the corresponding possible form of the kinetie energy density is thus given by[25] ... 

The above results are of great importance for two main reasons they allow us to regard the Wigner operators... 

Use the irrep set up in Problem 4.8 to define the Wigner operators p x (k, a = 1,2) that appear in (4.2.7). Verify that on applying them to the spin product 0 - afia they reproduce the brandling-diagram functions. 

Set up the irrep D5, dual to that obtained in Problem 4.8, and use it to form Wigner operators as in Problem 4.9. Apply a suitable pair of operators to a product a ri)b(r2)c(rx) to generate spatial functions that behave, under permutations of spatial variables, like the two basis vectors. Construct the function (4.3.5) and show that it is indeed emtisymmetric under space-spin permutations P12, P23, and hence satisfies the Pauli principle. 

To what linear combination of Slater determinants is the function obtained in Problem 4.11 equivalent Show that a second function may be obtained, by using an alternative peiir of Wigner operators, and hence that there are two linearly independent doublet functions (with S = M = ) for the configuration... 

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 ... 

The fact that, under the operations of the two groups, the functions are only mixed within the same row or column in (10.3.2) is a result of the commutation of permutation- and unitary-group elements (Problem 10.6). The behaviour of a row of functions, under unitary transformations, is thus unchanged if the functions are subjected to a common permutation. But the Wigner operators that we have used are simply linear combinations of permutations, and since leads from a function of species k to its partner of species jc, it follows that we may pass from the functions in row k of the array to those in row jc without affecting the way the... 

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