Operator electron annihilation

The operator annihilates a spin-up electron at p G A, and the operator b annihilates a spin-down electron The electrostatic operator... 

The action of operator a reduces nj from 1 to 0 if spin orbital i is occupied and gives zero if spin orbital i is unoccupied. The operators aj are for that reason called electron annihilation operators. A special case of Eq. (1.15) is... 

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. 

Electron annihilation operators, as Hermitian conjugates of creation operators, are no longer the components of an irreducible tensor. According to (13.40), such a tensor is formed by (21 + l)(2s + 1) components of the operator... 

For one shell of equivalent electrons representation (14.65) has been found in which the irreducible tensorial products are constructed separately for creation operators and annihilation operators. Likewise, such a representation can also be found for two-shell configurations. In that case,... 

We shall now introduce the electron creation operators a and b and the electron annihilation operators a = (—b = (—1 y mb for the electrons in the subshells nihjNl and n2hjNl, respectively. They are irreducible tensors of rank t = 1 /2... 

A iV-electron vacancy (hole) in a shell may be denoted as nl N = ni4l+2-N (see aiso Chapters 9, 13 and 16). As we have seen in the second-quantization representation, symmetry between electrons and vacancies has deep meaning. Indeed, the electron annihilation operator at the same time is the vacancy creation operator and vice versa instead of particle representation hole (quasiparticle) representation may be used, etc. It is interesting to notice that the shift of energy of an electron A due to creation of a vacancy B l is approximately (usually with the accuracy of a few per cent) equal to the shift of the energy of an electron B due to creation of a vacancy A l, i.e. 

The basic elements of the second-quantization formalism are the annihilation and creation operators (Linderberg and (3hrn, 1973). The annihilation operator ap annihilates an electron in orbital creation operator ap (the conjugate of ap) creates an electron in orbital p. These operators satisfy the anticommutation relations... 

Let Cy n) be the operator that annihilates an electron in the orbital state 3r/y at nth metal site. The ground-state energy level broadens into an energy band. The bandwidth is proportional to the effective overlap integral,... 

It is convenient to introduce an electron annihilation operator, written as a, which destroys an electron in spin orbital to give an (N — l)-electron ket if spin orbital , is occupied, and which gives zero if the orbital is unoccupied in the ket. For example... 

It may be readily verified that the matrix exp( - iA) is in fact unitary, provided the matrix A is Hermitian. The fact that the matrix exp( — iA) is unitary also means that the operator exp( — iA) is unitary and its matrix representation in the full ket expansion space, with matrix elements , is a unitary matrix. An analogous relation holds for transformations of the electron annihilation operators a, but it is the creation operator expansion that is most important for the MCSCF method. Substitution of the operator transformation into the expression of an arbitrary determinant gives the relation... 

The subscript i labels the principle quantum number and angular momentum quantum numbers (njlm). Here a,- and bj denote electron-annihilation and positron-creation operators, respectively, defined via diagonaiization of the unperturbed Hamiltonian (1.11)... 

Here, i CT(r) and (r) represent the ordinary nonrelativistic electron annihilation and creation operators. An LDA-type approximation has recently been derived for the exchange-correlation free energy Fxc[n, xl leading to explicit expressions for the effective potentials Veg(r) and Aeff (r, r ) (Kurth et al. 1999). 

In analogy with (18.18)—(18.20) we conclude that the operator Tf1 annihilates an electron in the first subshell and creates it in the second (and the operator Tvice versa), so that the total number of electrons in the subshells will remain unchanged (N = JVi + JV2 = N[ + N 2). The operator has the form i(N2 — N )/2, where N and N2 are the particle number operators for the first and second subshells, respectively. 

For ionization potentials, the customary EOM basis has the form O/ = a, a , a al,a, a ala ,...). These operators each annihilate one more electron than is created, thus producing an (A — l)-electron function when operating on an -electron function. The operator a removes an electron from an orbital that is occupied in reference determinant, and a al,a removes one electron from a hole orbital and excites a second electron from a hole orbital to a particle orbital. The operator oj, removes an electron from the particle orbital m and a ala removes an electron from a particle orbital and de-excites another electron from a particle orbital to a hole orbital. For electron affinities, the adjoints of the operators in the IP basis from the standard EOM operator basis. 

In this expression, a< and a] designate electron annihilation and creation operators, respectively. The projection operators in Eq. (71) are implemented by simply restricting the indices i, j, k, I to bound states and positive-energy continuum states, omitting contributions from negative-energy states entirely. In Eq. (74), the quantity Cj is the eigenvalue... 

As is done in the GL-theory for a single even-parity order parameter, we write the free energy density difference between the superconducting state and the norma state as an expansion in even powers of the complex gap function A(k), which is related to the anomalous thermal average of the microscopic theory [28] where c is the electron annihilation operator with wave vector k and spin t. However, for the multiple-order parameter case we must expand A(k) as a linear combination of the angular momentum basis functions Yj(k)),... 

A physical interpretation of the GF may now be obtained by considering the content of Eq. (6.6). If A and B are number conservir, (j.e., they both contain equal numbers of creation and aniii . ati " operators) then the states n> must contain the same number of electrons as the reference state 0> to give a nonvanishing GF. However, if A contains, for example, one more creation operator than annihilation operator, then n> must contain N 1 electrons (notice that the fact that the second-quantized H is independent of N is now becoming convenient). From the frequency spectrum of it is clear that the GF contains information about energy differ-... 

Before considering spin coupling details, we examine the Heisenberg equation of motion for the electron annihilation operators in the unperturbed case. Hubbard found that the two components nr-uttru and (1 — rir-v)arv of the... 

The operator cjin annihilates an electron in a local basis function centered in cell R. The sum in (5.66) is precisely the projector onto the irreducible representation k of the translation group. In this basis the matrix (5.65) becomes... 

With respect to the old vacuum, the new vacuum is called a dressed vacuum and the states in it are called dressed states. The new particles are called quasiparticles because they are a composite of an old particle and a series of old pairs they are dressed with a series of pairs rather than being bare particles. The new vacuum is also called a polarized vacuum because with respect to the old vacuum there is a charge polarization expressed by the presence of undressed pairs. The concept of vacuum polarization will be discussed more later. Finally, another way of looking at the change in the vacuum is that because the transformation mixes positron creation and electron annihilation operators, a normal-ordered operator in the new basis will definitely not be normal-ordered in the old basis. 

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