Annihilation operator effect

It is clear that if all the program counters arc unoccupied (i.e. are in the state I 0 >), nothing at all happens all terms in the Hamiltonian start out with an annihilation operator, and all states thus remain in the state 0 > for all time. If we assume that only one of the sites 0,1,... fc sites is occupied, however, we see that only one site will always be occupied, though not necessarily the same site at different times. If we think of the occupied site, say the first site 0, as a cursor, the Hamiltonian effectively moves the cursor along the program counter sites while the operators Ai operate on the register n. Feynman shows how, by the time the cursor arrives at the final site fc, the n register has been multiplied by the entire set of desired operators Tj, T2,..., A -... 

Here the indices a and b stand for the valence orbitals on the two atoms as before, n is a number operator, c+ and c are creation and annihilation operators, and cr is the spin index. The third and fourth terms in the parentheses effect electron exchange and are responsible for the bonding between the two atoms, while the last two terms stand for the Coulomb repulsion between electrons of opposite spin on the same orbital. As is common in tight binding theory, we assume that the two orbitals a and b are orthogonal we shall correct for this neglect of overlap later. The coupling Vab can be taken as real we set Vab = P < 0. 

Here frs and (ri-l tM> are, respectively, elements of one-electron Dirac-Fock and antisymmetrized two-electron Coulomb-Breit interaction matrices over Dirac four-component spinors. The effect of the projection operators is now taken over by the normal ordering, denoted by the curly braces in (15), which requires annihilation operators to be moved to the right of creation operators as if all anticommutation relations vanish. The Fermi level is set at the top of the highest occupied positive-energy state, and the negative-energy states are ignored. 

An annihilation operator times the vacuum state still vanishes so the effect of an annihilation operator times an occupation number vector becomes... 

This relationship expresses the adjoint character of the annihilation operators. The effect of the time reversal operator on these tensors in the Fano-Racah phase convention is given by ... 

When an accurate wave function is calculated by some effective method, we can calculate an accurate annihilation rate with the use of an effective annihilation operator [5] ... 

In the electronic Hamiltonian t +i, is the transfer integral, i.e. the re-electron wavefunction overlap between nearest neighbour sites in the polymer chain, and is equivalent to the parameter /3 in Equation (4.20), and c 1+,s. and clhS are creation and annihilation operators that create an electron of spin s ( 1/2) on the carbon atom at site n-f 1 and destroy an electron of spin, s at the carbon atom on site n, i.e. in effect transfer an electron between adjacent carbon atoms in the polymer chain. The elastic term is just the energy of a spring of force constant k extended by an amount ( +1— u ), where the u are the displacements along the chain axis of the carbon atoms from their positions in the equal bond length structure, as indicated in Fig. 9.8(b). The extent of the overlap of 7i-electron wavefunction will depend on the separation of nearest neighbour carbon atoms and is approximated by ... 

Here k is the momentum of the quasi-free electrons, whose single-particle energies include the effect of electron-electron repulsion renormalization and is the occupation number operator for state k). In Eq. (6.94) the electronic coupling between electrode and redox center is included which is governed by the matrix element between states k> and

cj and are the creation operators for the states in the redox system and the metal, respectively, whereas c and are the corresponding annihilation operators. Creation and annihilation means that an orbital k in the metal becomes occupied by an electron or emptied, respectively here in the presence of the electronic coupling. 

The representation of a staircase or a molecular chain in a BDS is very general, and embraces a family of recognizably similar systems. Operations on the BDS can effect transformations within this extensive set. Such operations fall into several distinct classes rigid-body motions, cylinder deformations, non-conservative transformations, and creation-annihilation operations. 

In summary, there are three different lanes available for evaluating the operator products on the rhs of the generalized Bloch Eq. (23) and for all effective operators, such as the Hamiltonian (15) or any other interaction operator. These alternatives are displayed in Figure 10.2 beside of the (i) purely algebraic transformation, i.e. the use of the fermion anticommutation rules for the creation and annihilation operators, variety of graphical methods and rules... 

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

Evaluation of the operator products as they occur either on the rhs of fhe Bloch Eq. (23) or in fhe definifion of fhe effective operators (15) and (49). The aim of this step is to bring all the creation and annihilation operators (in each term of the expansions) into the extended normal-order form (50). The resulf is a sequence of normal-ordered operator ferms (briefly referred to as Feynman-Goldstone diagrams). 

In the last few years the general theory of effective Hamiltonians has been reformulated by Kutzelnigg in the Fock space. The use of creation and annihilation operators is supposed to simplify the calculation of quantities involving variation of the number of particles, such as ionization potentials or electron affinities. Up to now no specific applications have been published with this formalism, the practical efficiency of which is still to be established. 

But it must be clear that this reduction of information and this focus on some low part of the spectrum proceed differently and lead to completely different tools. The effective Hamiltonians appear as N-electron operators acting in well defined finite bases of iV-electron functions. The effective Hamiltonians obtained from the exact bielectronic Hamiltonian introduce three- and four-body interactions. They may essentially be expressed as numbers multiplied by products of creation and annihilation operators. In contrast, the pseudo-Hamiltonians keep an a priori defined analytic form, sometimes simpler than the exact Hamiltonian to mimic. For instance, the... 

Here aj(aj) are creation and annihilation operators of the bath mode of frequency ojj and bc(bt) are annihilation operators of a coupling and a tuning mode, respectively. The so-called rotating-wave approximation (RWA) has been invoked in Eq. (17), neglecting terms of the type ba and b a. The effect of the bath on the system dynamics is completely determined by the... 

In spite of considering two-photon processes, we still find the energy levels (8.17) to be equidistant with respect to the photon occupation numbers n, -I- i. This suggests that it is permissible to neglect the effect of the electron-photon interaction on statistics and to occupy the electron states and the photon states according to Fermi statistics and to Bose statistics. We shall check this question in Section 8.4 by studying the creation and annihilation operators of the resulting quasi-electrons and quasi-photons. 

As emphasized above, we did not specify so far any direct connection between the creation operators aj and the annihilation operators aj. The only property required for aj was that it should annihilate the effect of in the sense ... 

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