Products of creation and annihilation operators

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

Muda and Hanawa (MH) have approached the problem by considering the time variation of the quantities (r) = < r) cl,c, t)>, which are expectation values of products of creation and annihilation operators for site-centered orbitals >. The Schrodinger equation then leads to a set of first-order differential equations, viz.,... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

Using second-quantization, it is often necessary to transform complicated tensorial products of creation and annihilation operators. If, to this end, conventional anticommutation relations (14.19) are used, then one proceeds as follows write the irreducible tensorial products in explicit form in terms of the sum over the projection parameters of conventional products of creation and annihilation operators, then place these operators in the required order, and finally sum the resultant expression again over the projection parameters. On the other hand, the use of (14.21) enables the irreducible tensorial products of second-quantization operators to be transformed directly. 

It has been shown in the previous section how the submatrix elements of irreducible tensorial products of creation and annihilation operators can be expressed in terms of pertinent one-shell submatrix elements. The submatrix elements of the operators G1-G7 are also defined in terms of the same quantities. Since the quasispin ranks from different shells that... 

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,... 

These boson operators are related to those introduced in [20] as shown in Table 7. They transform under D4/, as the representation indicated by the appropriate letter. [An extended s-wave i can also be introduced.] One then constructs the Hamiltonian by expanding it into bilinear products of creation and annihilation operators, with the constraint that H must transform as the representation of... 

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 ... 

To proceed further, we have to know how to handle the products of creation and annihilation operators. It is Wick s theorem which tells us how to deal with the products of these operators. Before presenting Wick s theorem we have to introduce some necessary definitions and relations. The creation and annihilation operators satisfy the anticommutation relation... 

Expressed in words, the product of creation and annihilation operators is equal to a normal product of these operators plus the sum of normal products with one contraction plus the sum of normal products with two contractions etc., up to the normal product where all operators are contracted. This theorem can be generalized (to the so-called generalized Wick s theorem) in the way that also a product of the following form can be handled ... 

In many-body problems we are normally concerned with systems that have a fixed number of electrons. In order to preserve the electron number N we must operate on the configuration p) with equal numbers of creation and annihilation operators. We now develop the algebra of these operators. First we consider products of creation and annihilation operators for one orbital a). 

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... 

We expanded the X operators into the primitive products of creation and annihilation operators to achieve this factorization. Instead of first expanding the x in terms of X, we can expand them directly in the primitive products of creation and annihilation operators, to obtain the sums of products of parentage coefficients. As an example, consider the second off-diagonal block, for which the two-particle density matrix factorizes naturally ... 

In eq. (3.91), the notation is such that A and C (B and D) have the same space and spin coordinates. We emphasize again that the sequence of the creation and annihilation operators in the second term of eq. (3.91) is very important. We note that in the second quantized Hamiltonian, IK, defined in eq. (3.91), we have products of creation and annihilation operators. However, note that the Hamiltonian operator in this form does not depend on the number of particles. It is completely defined by the one-electron functions A), B).... ... 

To make further progress, we must understand how to handle products of creation and annihilation operators of the type arising in eq. (3.91). We shall introduce Wick s theorem and also some necessary definitions and relations. 

Having defined both a normal product of creation and annihilation operators and the contraction of a pair of these operators, we are now in a position to define a normal product with contractions. Consider, for example, the following simple case ... 

Wick s theorem (3.109), which gives the prescription for handling products of creation and annihilation operators may, of course, be applied to the Hamiltonian operator, 5f, when it is expressed in the second-quantization formalism, eq. (3.91) ... 

In these equations, f is the Hartree-Fock operator and X (Xg) are creation (annihilation) operators defined with respect to Fermi vacuum o) and N[ ] is the normal ordered product of creation and annihilation operators. 

The algebraic manipulation of products of creation and annihilation operators is greatly simplified by the time-independent Wick s theorem [1,2],... 

Any product of creation and annihilation operators can be written as the sum of the normal product plus all possible singly, multiply and fully contracted normal products. 

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