Wave functions in quasispin space

It follows from (14.5) that when Q operates on wave function (15.44), this changes only the z-projection of the quasispin momentum, and so we can obtain from the wave function of N electrons appropriate values for N + 2 electrons with the same quantum numbers 

Since the wave functions with N v can be found from the wave functions with N = v using (16.1), in the second-quantization representation it is necessary to construct in an explicit form only wave functions with the number of electrons minimal for given v, i.e. IolQLSMq = —Q). But even such wave functions cannot be found by generalizing directly relation (15.4) if operator cp is still defined so that it would be an irreducible tensor in quasispin space, then the wave function it produces in the general case will not be characterized by some value of quantum number Q v). This is because the vacuum state 0) in quasispin space of one shell is not a scalar, but a component of a tensor of rank Q = l + 1/2 

All the other components of this tensor - the wave functions of Js with different numbers of electrons - can be obtained from (16.1). 

Operator q (lN)av LS in the general case consists of a linear combination of creation operators that provides a classification of states according to quantum numbers a, v. Since each term of this expansion contains N creation operators then from (15.49) the quasispin rank of operator p(/Wyxt (LS) an(j jts projectjon are equal to N/2 (regardless of the values of quantum numbers a, v). Accordingly, for a function with a certain 

The rank of the operator q in quasispin space will only be determined by the number of electrons N and can be ignored as a characteristic. In particular, operator - a tensor of rank one in quasispin space - acting on the vacuum state gives rise to the two-electron wave function with v = 0 

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It has been shown earlier (see Chapters 15 and 16) that the technique relying on the tensorial properties of operators and wave functions in quasispin, orbital and spin spaces is an alternative but more convenient one than the method of higher-rank groups. It is more convenient not only for classification of states, but also for theoretical studies of interactions in equivalent electron configurations. The results of this chapter show that the above is true of more complex configurations as well. 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

The wave function of the two-shell configuration (17.42) corresponds to the representation of uncoupled quasispin momenta of individual shells. The eigenfunction of the square of the operator of total quasispin and its z-projection can be written as follows in the scheme of the vectorial coupling of momenta in quasispin space ... 

These tensorial products have certain ranks with respect to the total orbital, spin and quasispin angular momenta of both shells. We shall now proceed to compute the matrix elements of such tensors relative to multi-configuration wave functions defined according to (17.56). Applying the Wigner-Eckart theorem in quasispin space to the submatrix element of tensor gives... 

As in the case of LS coupling, the tensorial properties of wave functions and second-quantization operators in quasispin space enable us to separate, using the Wigner-Eckart theorem, the dependence of the submatrix elements on the number of electrons in the subshell into the Clebsch-Gordan coefficient. If then we use the relation of the submatrix element of the creation operator to the CFP... 

There is another way of looking at the tensorial properties of operators and wave functions in the quasispin space of the entire two-shell configuration. If now we introduce the basis tensors for two subshells of equivalent electrons... 

Thus, the use of wave function in the form (23.67) and operators in the form (23.62)—(23.66) makes it possible to separate the dependence of multi-configuration matrix elements on the total number of electrons using the Wigner-Eckart theorem, and to regard this form of superposition-of-configuration approximation itself as a one-configuration approximation in the space of total quasispin angular momentum. 

The use of the tensorial properties of both the operators and wave functions in the three (orbital, spin and quasispin) spaces leads to a new very efficient version of the theory of the spectra of many-electron atoms and ions. It is also developed for the relativistic approach. 

As has been shown, second-quantized operators can be expanded in terms of triple tensors in the spaces of orbital, spin and quasispin angular momenta. The wave functions of a shell of equivalent electrons (15.46) are also classified using the quantum numbers L, S, Q, Ml, Ms, Mq of the three commuting angular momenta. Therefore, we can apply the Wigner-Eckart theorem (5.15) in all three spaces to the matrix elements of any irreducible triple tensorial operator T(JC K) defined relative to wave functions (15.46)... 

The relationships describing the tensorial properties of wave functions, second-quantization operators and matrix elements in the space of total angular momentum J can readily be obtained by the use of the results of Chapters 14 and 15 with the more or less trivial replacement of the ranks of the tensors l and s by j and the corresponding replacement of various factors and 3nj-coefficients. Therefore, we shall only give a sketch of the uses of the quasispin method for jj coupling, following mainly the works [30, 167, 168]. For a subshell of equivalent electrons, the creation and annihilation operators a and a(jf are the components of the same tensor of rank q = 1/2... 

Here (P)0 7 ) is the second-quantized operator producing the relevant normalized wave function out of vacuum, i.e. it simply generalizes the relationship (15.4) to the case of tensors with an additional (isospin) rank. Since the vacuum state is a scalar in isospin space (unlike quasispin space), the expressions for wave functions and matrix elements of standard quantities in the spaces of orbital and spin momenta can readily be generalized by the addition of a third (isospin) rank to two ranks in appropriate formulas of Chapter 15. 

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