Creation/annihilation

The following angle-action representations [14] are available for the classical equivalents of the right-hand and left-hand creation-annihilation operators [46] ... 

As before, we can perform reverse simulations. Instead of annihilating the solute, we can create it by turning on the perturbation part of the Hamiltonian. The resulting free energy differences are connected through the relation Z A reation — creation = Annihilation - annihilation- Comparison of this creation scheme with the transformation described by the horizonal arrow reveals two important differences. First, the vertical transformations require two sets of simulations instead of one, although one of them involves only solute in the gas phase and, is, therefore, much less computationally intensive. Second, the two methods differ in their description of the solute in the reference state. In both cases the solute does not interact with the solvent. For the vertical transformations, however, all interactions between atoms forming the solute vanish, whereas in the horizontal transformation, the molecule remains intact. 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

Positive semidefinite Hamiltonians can be constructed by taking an operator that depends on g creation/annihilation operators and multiplying the operator... 

Equations (1.206) and (1.207) describe the ionization of neutral vacancies (Vx, Vm). We assume here that the ionization of V and Vm to Vx and Vm does not take place. In a crystal in thermal equilibrium, electrons and holes will be formed by thermal excitation of electrons from the valence band to the conduction band, and the reverse process is also possible. This process can be expressed by eqn (1.210) as a chemical reaction, (see eqn (1.136)). Such reactions are called creation-annihilation reactions. Equations (1.208) and (1.209) describe the creation-annihilation reactions of neutral vacancies and charged vacancies in a crystal. Equation (1.211) shows the formation reaction of MX from constituent gases. It is to be noted that of these eight equations two are not independent. For example, the equilibrium constants Ks and K x in eqns (1.209) and (1.211) are expressed in terms of the other Ks as... 

Bose birth-death (creation-annihilation) operators a-, at, 6-, fet ... 

The three curves shown in Fig. 3 are the ones calculated by using this Hamiltonian. Here, f J1 is the electronic transfer T between the fiu orbital a of the mth C60 molecule and the fiu orbital b of the nth C o molecule, where a and b denote x, y, and z f is chosen so as to reproduce the result of the electronic structure calculations. We also use spin electron in the flu orbital a of the mth C60 molecule. Furthermore, is the band energy of the flu electron of the band index a (a = 1,2, and 3) and the wavenumber k the band energies are obtained by diagonalizing the Hamiltonian H0 and we use ak(J(akli) to denote the corresponding creation (annihilation) operators. 

In terms of the creation-annihilation electron and phonon operators the Hamiltonian can be cast as follows ... 

As usual, the index k includes the wave vector k and the polarizations e [ and e2 perpendicular to k, and al (at) is the creation (annihilation) operator for photons of energy h(ok = he k 1. We must write in this formalism the operators, vector potential, and electric field which are involved in our calculations ... 

In this subsection we describe a discrete model for vacancy-mediated diffusion of embedded atoms, solve it numerically for the case of In/Cu(0 0 1), and present the results. Our model is defined on a two-dimensional simple square lattice of size / x / (typically, l = 401) centered around the origin. This corresponds to the top layer of a terrace of the Cu(00 1) surface, with borders representing steps. The role of steps in the creation/annihilation of vacancies will be discussed in more detail in the next section. All sites but two are occupied by substrate atoms. At zero time the two remaining sites are the impurity (or tracer) atom, located at the origin, and a vacancy at position (1,0). This corresponds to the situation immediately after the impurity atom has changed places with the vacancy. 

For the analysis of the various formalisms, manipulation of the equations, generating normal product of terms via Wick s theorem, and particularly for indicating how the proofs of the several different linked cluster theorems are achieved, we shall make frequent use of diagrams. For the sake of uniformity, we shall mostly adhere to the Hugenholtz convention/1/. All the constituents of the diagrams will be operators in normal order with respect to suitable closed-shell determinant taken as the vacuum. We shall refer to the creation/annihilation operators with respect to this vacuum after the h-p transformation.The hamiltonian H will also be taken to be in normal order with respect to... 

The energy of the reaction again does not contain any contribution from the Si—H bond energy, because the bond is preserved. Furthermore, the two defects can diffuse apart via reaction (6.91). Equivalent reactions can be written for the creation, annihilation and migration of dopant states. 

The creation operators in Eqs. [22] and [23] are restricted to act only on the virtual orbitals, and the annihilation operators may act only on the occupied orbitals. Therefore, by Eq. [21], the creation-annihilation operator pairs exactly anticommute ... 

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