Creation operators spin properties

We shall gradually construct the formalism of second quantization by showing how the properties of determinants can be transferred onto the algebraic properties of operators. We begin by associating a creation operator aj with each spin orbital We define aj by its action on an arbitrary Slater... 

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 creation operators aj, for nonorthogonal spin orbitals are defined in the same way as for orthonormal spin orbitals (1.2.5). As for orthonormal spin orbitals, the anticommutation relations of the creation operators and the properties of their Hermitian adjoints (the annihilation operators) may be deduced from the definition of the creation operators and from the inner product (1.9.2). However, it is easier to proceed in the following manner. We introduce an auxiliary set of symmetrically orthonormalized spin orbitals... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

In turn, the monochromatic multipole photons are described by the scalar wavenumber k (energy), parity (type of radiation either electric or magnetic), angular momentum j 1,2,..., and projection m = —j,..., / [2,26,27]. This means that even in the simplest case of monochromatic dipole (j = 1) photons of either type, there are three independent creation or annihilation operators labeled by the index m = 0, 1. Thus, the representation of multipole photons has much physical properties in comparison with the plane waves of photons. For example, the third spin state is allowed in this case and therefore the quantum multipole radiation is specified by three different polarizations, two transversal and one longitudinal (with respect to the radial direction from the source) [27,28], In contrast to the plane waves of photons, the projection of spin is not a quantum number in the case of multipole photons. Therefore, the polarization is not a global characteristic of the multipole radiation but changes with distance from the source [22],... 

The polarization and quantum phase properties of multipole photons change with the distance from the source. This dependence can be adequately described with the aid of the local representation of the photon operators proposed in Ref. 91 and discussed in Section V.D. In this representation, the photon operators of creation and annihilation correspond to the states with given spin (polarization) at any point. This representation may be useful in the quantum near-field optics. As we know, so far near-field optics is based mainly on the classical picture of the field [106]. 

Although Eqs. (1.2)-(1.5) contain all of the fundamental properties of the Fermion (electron) creation and annihilation operators, it may be useful to make a few additional remarks about how these operators are used in subsequent applications. In treating perturbative expansions of N-electron wavefunctions or when attempting to optimize the spin-orbitals appearing in such wavefunctions, it is often convenient to refer to Slater determinants that have been obtained from some reference determinant by replacing certain spin-orbitals by other spin orbitals. In terms of second-quantized operators, these spin-orbital replacements will be achieved by using the replacement operator as in Eq. (1.9). 

Thus, all pairs of creation and/or annihilation operators anticommute except for the conjugate pairs of operators such as ap and ap. From these relationships, all other properties of the creation and annihilation operators - often referred to as the elementary operators of second quantization - follow. We note that equation (103) holds only for orthonormal sets of spin orbitals. For nonorthonormal spin orbitals, the Kronecker delta in equation (103) must be replaced by the overlap integral between the two spin orbitals. 

What combination of Fock-space operators will produce, when acting on a closed-shell state vector o)> a resultant vector equivalent to the wavefunction used in Problem 8.1 Use the second-quantization form of the Hamiltonian (p. 82) and the anticommutation properties of the creation and annihilation operators (p. 81) to give an alternative derivation of the energy expression found in Problem 8.1. Hint Use operators a , etc. to destroy or create up-spin or down-spin electrons in orbital ipr- It is convenient to use indices i, j,... for orbitals in 0o and m, n,... for the virtual set used in the O.]... 

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