Concept of Creation and Annihilation Operators

This section is devoted to describe the second quantized representation of simple wave functions. To begin with, we introduce some general concepts characteristic to the theory of second quantization. 

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Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

The concept of quasispin quantum number was discussed in the Introduction and Chapter 9 (see formulas (9.22) and (9.23)). Now let us consider it in the framework of the second-quantization technique. We can introduce the following bilinear combinations of creation and annihilation operators obeying commutation relations (14.2) - the quasispin operator ... 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

Before considering the evaluation and simplihcadon of commutators and anticommutators, it is useful to introduce the concepts of operator rank and rank reduction. The (particle) rank of a string of creation and annihilation operators is simply the number of elementary operators divided by 2. For example, Ihe rank of a creation operator is 1/2 and the rank of an ON operator is 1. Rank reduction is said to occur when the rank of a commutator or anticommutator is lower than the combined rank of the operators commuted or anticommuted. Consider the basic anticommutation relation... 

X, xAa1 6 G and , a1 [f 6 G. The interior multiplications a11 and J a, decrease the rank of tensors in G and G, respectively. It is obvious that the exterior multiplications a A and A a1 and the interior multiplications a1 [ and J a may be formalized as the familiar creation and annihilation operators on the direct V i and adjoint i spaces, respectively, except that now the operators and their adjoints do not have the same domain. Specifically (a A)t =Jai. These concepts are part of the general theory of Dual Grassmann Algebras /67/. Some other relevant results are ... 

The radiation field is an ensemble of photons with an energy of ho) (o) = c A ) and a momentum of hk. By quantizing the electromagnetic field we have been able to introduce the concept of a photon (number). The vector potential can be rewritten as Eq. (1.108) by using creation and annihilation operators. 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

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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