Second quantization formalism state vector

The occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

The notation concerns are easily overcome by the following simple construct bearing the name of second quantization formalism.21 Let us consider the space of wave functions of all possible numbers of electrons and complement it by a wave function of no electrons and call the latter the vacuum state vac). This is obviously the direct sum of subspaces each corresponding to a specific number of electrons. It is called the Fock space. The Slater determinants eq. (1.137) entering the expansion eq. (1.138) of the exact wave function are uniquely characterized by subsets of spin-orbitals K = k,, k2,..., fc/v which are occupied (filled) in each given Slater determinant. The states in the list are the vectors in the carrier space of spin-orbitals (linear combinations of the functions of the (pk (x) = ma (r, s) basis. We can think about the linear combinations of all Slater determinants, may be of different numbers of electrons, as elements of the Fock space spanned by the basis states including the vacuum one. 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

Second quantization treats operators and wave functions in a unified way - they are all expressed in terms of the elementary creation and annihilation operators. This property of the second-quantization formalism can, for example, be exploited to express modifications to the wave function as changes in the operators. To illustrate the unified description of states and operators afforded by second quantization, we note that any ON vector may be written compactly as a string of creation operators working on the vacuum state (1.2.4)... 

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. 

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See also in sourсe #XX -- [ ]

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