Pair creation and annihilation

In the present paper, we shall discuss a method for generating many-electron states of a given symmetry using Kramers pair creation operators and other symmetry-preserving pair creation and annihilation operators. We will first develop the formalism for the case where orthonormality between the orbitals of different configurations can be assumed. Afterwards we will extend the method to cases where this orthonormality is lost, so that the method also can be used in generalized Sturmian calculations [11-13] and in valence bond calculations. 

Thus, the Kramers pair creation and annihilation operators defined by equations (36) and (37) preserve the symmetry of the states on which they act. 

Using the anticommutation relations (32), we can obtain the following commutation relations for the Kramers pair creation and annihilation operators [8] ... 

Soon afterward, other phenomena such as Compton scattering. X-ray production, pair creation and annihilation could be interpreted successfully using a photon picture of light. Light still retains its wavelike properties as it travels through space. It assumes its photon or particle-like behavior only when it interacts with matter in a detector or at a target. 

Quantum mechanical calculations in the molecular sciences do not necessarily involve a variation of the number of particles (especially not through pair creation and annihilation processes). This even holds true in the case of particle exchange processes as the reactants involved can be described in a fixed-particle-number framework. For example, a reductant can be treated together with the molecule to be reduced as a whole system such that the number of electrons remains constant during the reduction process. Also, the energy of liberated electrons can be considered zero, and thus such electrons can be neglected from one step to the next in a reaction sequence. This is, for instance, useful for ionization processes, where the released electron is considered to be at rest and features zero energy at infinite distance so that it makes no contribution to the Hamiltonian of the ionized system. There is therefore a need to proceed from QED to a computationally more appropriate albeit less... 

Figure 2.2 A gas of electrons and positrons in equilibrium with radiation at very high temperatures. At temperatures over 10 K, particle-antiparticle pair creation and annihilation begins to occur and the total number of particles is no longer a constant. At these temperatures, electrons, positrons and photons are in the state called thermal radiation. The energy density of thermal radiation depends only on the temperature...

When we consider interconversion of particles and radiation, as in the case of particle-antiparticle pair creation and annihilation, the chemical potential of thermal photons becomes more significant (Fig. 11.4). Consider thermal photons in equilibrium with electron-positron pairs ... 

We now introduce a notation involving a normal product with one or more contracted pairs of factors. If U, V, denote a set of free-field creation and annihilation operators, we define the mixed product by... 

Here AB is the difference in ionization potentials of AT and GC base pairs, b is the transfer integral, c, and c, are the creation and annihilation operators for a hole at the f-th site, respectively, index i labels DNA base pairs in the sequence, and the sum E is taken over GC sites only. 

All these formulas for a single pair of creation and annihilation operators obviously apply to a more general situation of dim R pairs. The matrix elements... 

All four creation and annihilation operators for electrons in the pairing state (a,/ ) can be expressed via the tensor in (15.38) at various values of the projections v and m. The anticommutation relations (18.8) and... 

Any products of creation and annihilation operators for electrons in a pairing state can be expanded in terms of irreducible tensors in the space of quasispin and isospin. So, for the operators (18.10) and (15.35) we have, respectively,... 

There is a second, alternative approach. One could assume that the unpaired neutron and the unpaired proton form a quasibound state. The total number of components of the angular momenta of this quasi-bound state is given by n n v. Then we introduce a pair of new bosonic creation and annihilation operators associated with each level of this subsystem, cj, Cj, I,J =... 

We now introduce creation and annihilation operators ajj and an which create/annihilate e-h pairs at a given combination of sites n = (n, n1), i.e., 41°) = 14 = nen h), where 0) is the ground state. Using these operators, a generic monoexcitation configuration interaction Hamiltonian can be formulated as follows in second quantization notation,... 

The narrow stripon band splits in the SC state, through the Bogoliubov transformation, into the EL(ft) and +(k) bands, given in Eq. (20). The states in these bands are created, respectively, by pl(k) and p+(k), which are expressed in terms of creation and annihilation operators of stripons of the two pairing subsets [see Eq. (16)] through equations of the form ... 

There is another consequence of the photon cloud around an electron. In this cloud of photons, the creation and annihilation of particles occur. It is these virtual particles, pairs of positive and negative particles, that lead to the polarization of the empty space... 

For spin-orbitals, the creation and annihilation operator pair is a generator of the unitary group, and so Eq. (52) can be written... 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

Next, insert the resolution of the identity between the pairs of creation and annihilation operators in the two-electron term. Clearly, the sum must run over (N — 2)-electron states. The expression for a becomes... 

Garavelli, M., Smith, B. R., Bearpark, M. J., Bemardi, F., Olivucci, M., Robb, M. A., Relaxation Paths and Dynamics of Photoexcited Polyene Chains Evidence for Creation and Annihilation of Neutral Soliton Pairs, J. Am. Chem. Soc. 2000,122, 5568 5581. 

In (1) a ic.b Kjaic and b f are boson creation and annihilation operators for the a and b Hartree-Fock particle states with momentum hn and kinetic energy K-h k /2m. pa and pb denote the densities of the component holon gases while pa and pb denote their respective chemical potentials. In equilibrium, pa=Pb-Pt where u is determined by the condition that the statistical average of A Eic (a icaic b Kbic) be equal to p=pa+pb=N/A, the total number of holons per unit area (N , A ). V is taken to satisfy V<pairing interaction. Finally, V is restricted to operate between holons with k[Pg.45]

In second quantization, the numerical vector-coupling coefficients (the Aff and ) appear as matrix elements of creation and annihilation operators X] and jc> The operator X creates an electron in an orthonormal spin orbital io), where /(j) = /) (j), and (T = a or p. Similarly, operator destroys an electron in the orthonormal spin orbital ia). In quantum chemistry problems in which the number of particles is conserved, the Xj and will always occur in pairs. The role of these operators is easily illustrated by showing their operation on a specific type of CSF, namely a Slater determinant. Thus, as an example, for the determinant... 

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