Commutation relations. Occupation numbers

It should be stressed that in the literature one can come across a wide variety of notations for creation and annihilation operators. In this book we follow the authors [14, 95] who attach the sign of Hermitian conjugation to the electron annihilation operator, but not to the electron creation operator. Although the opposite notation is currently in wide use, it is inconvenient in the theory of the atom, since it is at variance with the common definitions of irreducible tensorial quantities. 

Let us now turn to the commutation relations between second-quantization operators. Acting in succession with two different creation operators on a one-determinant wave function, from (13.2), we get 

Since these two equations hold for any one-determinant wave function, and the functions on the right side of these equations only differ in sign, we arrive at the following anticommutation relation for the creation operators  

Subjecting this to Hermitian conjugation, we find that the same anticommutation relation is also obeyed by the annihilation operators 

From these anticommutation relations, specifically, we obtain the relation 

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The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

The effect of inter-site hopping is then introduced into the system. The manifold of basis states are limited to those in which the local correlations have been diagonalized. The wave functions for the composite particles then obey Bloch s theorem, which results in the formation of a dispersion relation consisting of two bands for the quasi-bosons the first band describes spinless quasi-boson excitations, the second band describes the magnetic quasi-bosons. Although these composite particles are bosons in that they commute on different sites, they nevertheless have local occupation numbers which are Fermi-Dirac like. 

This relation follows from the observation that the spin-projection operator (2.2.30) is a linear combination of spin-orbital ON operators. Thus, since the ON operators commute among themselves (1.3.4), they must also commute with the spin-projection operator. From the commutation relations (2.4.5) and from the observation that there are no degeneracies among the spin-orbital occupation numbers (2.4.4), we conclude that the Slater determinants are eigenfunctions of the projected spin ... 

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