One-particle operators of physical quantities

In Chapter 19 a number of operators corresponding to physical quantities will be expressed in coordinate representation in terms of irreducible tensors. In the most general form, the tensorial structure of one-electron operator / can be written as follows  

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 

Specifically, the operators of kinetic and potential energy of electrons (see (1.15)) are scalars in orbital and spin spaces. Therefore, their sum is at once found to be for a shell nlN 

Hi = N nlmp Hi nlmp) H = N(nlmp H nlmp). (14.54) A more complex tensorial structure is inherent in the operator of the 

This operator, too, is a scalar in the space of total angular momentum for an electron. Tensors in this space are, for example, the operators of electric and magnetic multipole transitions (4.12), (4.13), (4.16). So, the operator of electric multipole transition (4.12) in the second-quantization representation is 

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In Chapter 14 we have shown how an expansion in terms of irreducible tensors in the spaces of orbital and spin angular momenta for one shell can be obtained for the operators corresponding to physical quantities. The tensors introduced above enable the terms of a similar expansion to be also defined in the space of a two-shell configuration. So, for the one-particle operator of the most general tensorial structure (14.51) we find, instead of (14.52),... 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

Similar data for the dN configuration may be found in [91]. Applying such tables we can directly find the matrix elements of one-particle operators corresponding to physical quantities. The matrix elements of two-particle... 

Operators corresponding to physical quantities can also be expanded in terms of irreducible tensors in the quasispin space of each individual shell. To this end, it is sufficient to go over to tensors (17.43) and next to provide their direct product in the quasispin space of individual shells. This procedure can conveniently be carried out for a representation of operators such that the orbital and spin ranks of all the one-shell tensors are coupled directly. Here we shall provide the final result for the two-particle operator of general form (14.57)... 

With this brief discussion of the three basic steps (a)-(c) from Subsecfion 4.1, we have arrived at our original destination to represent all physical quantities of interest as sum (of products) of one- and two-parficle amplifudes. In practice, each of these expansions are often lengthy and the complexity of fhese expansions increases rapidly if the number of particle and holes is increased in the valence shells. The latter can be seen easily from the fact that each (valence-shell) particle of hole introduces an additional creation or annihilation operator into the operator strings a)a and (a ), respectively. In contrast to other, e.g. multiconfigurational, expansions of the wave functions, however, the explicit form of the approximate states IV a) cannot be derived so easily in MBPT or the CCA. For this reason also, a straightforward and simply handling of the perturbation expansions decides how successfully the theory can be applied to open-shell atoms and molecules in the future. 

This postulate pertains to ideal measurements, i.e. such that no error is introduced through imperfections in the measurement apparatus. We assume the measurement of the physical quantity A, represented by its time-independent operator A and, for the sake of simplicity, that the system is composed of a single particle (with one variable only). 

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