Operators and wave functions in second-quantization representation

4 Operators and wave functions in second-quantization representation 

Many-electron wave functions in second-quantization form can conveniently be represented in an operator form. To this end, we shall introduce the vacuum state 0), i.e. the state in which there are no particles. We shall define it by 

In principle, an A-electron wave function can always be represented as a certain combination of creation operators acting on the vacuum state. Specifically, for the one-determinant wave function (13.1) we have 

We shall use operators with the sign only when the same symbols are used both for operator and non-operator quantities. 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are 

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