Second-quantization. Electron creation and annihilation operators

2 Second-quantization. Electron creation and annihilation operators 

In the second-quantization method it is convenient to use the following designations of the JV-electron function instead of (10.9), often omitting the subscripts of coordinates, if this does not lead to any confusion  

Wave functions (13.1) form an orthonormal set, but their normalization factors are defined only up to a sign. The fact is that the wave function (13.1) is antisymmetric not only under coordinate permutations, but also under permutations of one-electron quantum numbers. Thus, to fix the sign of a wave function requires a way of ordering the set of quantum numbers (a) = ai, 0C2. ajy. There exists, however, a convenient formalism that allows us to include the constraints imposed by the requirement that the wave functions be antisymmetric in a simple operator form. This formalism became known as the second-quantization method. This chapter gives a detailed description of the fundamentals of the second-quantization method. 

The electron creation operator is introduced in such a way that the one-electron wave function produced by it would appear in the first line of the determinant with the same sign. Thereby the sign of the one-determinant wave function is uniquely specified also in the case where the resultant one-particle wave function is in any line of the determinant 

Let us look at operator at, which is the Hermitian conjugate of the creation operator. Using the definition of the Hermitian conjugated operator 

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