Tensors in quasispin space

Consequently, the quantum numbers of quasispin Q and seniority v are related by expression (9.22) which is also valid for the wave function in the general case at N v. Operator Qj1, acting on wave function (15.44), increases the number of particles by two paired electrons, leaving, by definition, number v unchanged. 

In summary, there exists a one-to-one correspondence between quantum numbers Q, Mq and v, N that is described by (9.22) and (15.42). The wave function will now be (9.25) or 

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 

It follows from (15.49) that the tensors aiqls have the following property with respect to the operation of Hermitian conjugation  

Anticommutation relations (14.19) can be rewritten for the components of triple tensors  

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We have established above that one-electron operators are expressible in terms of tensors W(kK) related to triple tensors W KkK by (15.59). Therefore, we shall find here the expansion in terms of irreducible tensors in quasispin space only for the two-particle operator that is a scalar in the total momentum. 

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... 

Rotation of any irreducible tensor in quasispin space transforms it according to the D-matrix of relevant dimensionality. For example, unitary transformations of a(qls yield the creation operators... 

Utilizing the tensorial properties of operators in quasispin space, we can, in particular, find an expansion of the scalar products of the operators Tk in terms of irreducible tensors in quasispin space... 

For operators corresponding to physical quantities, we can also obtain an expansion in terms of irreducible tensors in quasispin space. Specifically, for two-particle operators (13.23) that are scalars with respect to the total momentum J... 

Ayl transforms as an irreducible tensor operator under operations of G, and as a rank-2 spinor in the angular momentum algebra generated by the quasispin operators. We form the quasispin generators as a coupled tensor in quasispin space Q(A) = i[AAAA]7V2, where [AB] = Y.qq lm q c/)AqBqi. In the Condon and Shortley spherical basis choice (with m = 1, 0, — 1) for the SO(3) Clebsch-Gordan coefficients [11-13,21-23] this takes the form [6,21] ... 

By computing the commutators of the components of the quasispin operator with electron creation and annihilation operators, we can directly see that the latter behave as the components of a tensor of rank q = 1/2 in quasispin space and obey the relationship of the type (14.2)... 

Let us now return to the Casimir operators for groups Spy+2, SU21+1, R21+1, which can also be expressed in terms of linear combinations of irreducible tensorial products of triple tensors WiKkK To this end, we insert into the scalar products of operators Uk (or Vkl), their expressions in terms of triple tensors (15.60) and then expand the direct product in terms of irreducible components in quasispin space. As a result, we arrive at... 

It is worth mentioning that expressions (15.75) and (15.76), apart from terms that are scalar with respect to quasispin, also include terms that contain a second-rank tensor part in quasispin space. Substituting (15.75) and (15.76) into the expressions for the Casimir operators of appropriate groups (5.34), (5.29) and (5.33) yields... 

In Chapter 14 we derived, in the second-quantization representation, two different forms of the expressions for that operator - (14.61) and (14.63). To begin with, we consider expression (14.63) in which we, by (15.49), go over to triple tensors. Then, after some transformations and coupling the momenta in quasispin space, we arrive at... 

In summary, the techniques described in this chapter allow us to derive expansions of the operators that correspond to physical quantities, in terms of irreducible tensors in the spaces of orbital, spin and quasispin momenta, and also to separate terms that can be expressed by operators whose eigenvalues have simple analytical forms. Since the operators of physical quantities also contain terms for which this separation is impossible, the following chapter will be devoted to the general technique of finding the matrix elements of quantities under consideration. 

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... 

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)... 

Tensors in the space of total quasispin and their submatrix elements... 

Now let us represent the two-particle operator of general form (14.57) in terms of the irreducible tensors in the space of total quasispin for the two-shell configuration... 

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... 

Any products of creation and annihilation operators for electrons in a pairing state can be expanded in terms of irreducible tensors in the space of quasispin and isospin. So, for the operators (18.10) and (15.35) we have, respectively,... 

Considering the tensorial properties of the electron creation and annihilation operators in quasispin space, we shall introduce the double tensor... 

In (23.80) and (23.81) the rank sum y+k is an odd number, otherwise these operators are identically equal to zero. We shall separate sets of operators that are scalars in the space of total angular momentum but tensors in isospin space. If we go through a similar procedure for one subshell of equivalent electrons we shall end up with the quasispin classification of its states. It turns out that ten operators l/(00), U 0 vffl, F 0) are generators of a group of five-dimensional quasispin, wnich can be easily verified by comparing their commutation relations with the standard commutation relations for generators of that group. 

Now we shall have to express operators for physical quantities in terms of irreducible tensors in the spaces of total angular momentum and quasispin. One-electron terms of relativistic energy operator (2.1) (formulas (2.2)-(2.4)) are expressed in terms of operators (23.69), (23.71)-(23.73) in a trivial way. With two-electron operators the procedure of deriving the pertinent relations is more complex. The relativistic counterpart of (18.50)... 

Between states of well-defined quasispin Q, <2, the Wigner-Eckart theorem in quasispin space shows that any quasispin tensor has matrix elements proportional to... 

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