Secondary quantization and standard values of the theory of atomic spectra in j-representation

Abstract

Two binary operators and formed by the tensor product of the operators of creation and annihilation of electrons are introduced. The operator A+ is obtained by the tensor product of two operators of electron creation and, in a particular case, transforms into the raising operator Q+ of the quasi-spin. The operator B is obtained by the tensor product of the creation and annihilation operator of the electron and, in a particular case, changes to the operator of the number of electrons. Various commutation properties of the binary operators A+ and B are investigated. In a specific situation, they transform into commutation relations for the operators of quasi-spin and the number of electrons. Relationships have been established between the matrix elements of these operators concerning the state functions with different numbers of equivalent electrons. The following has been established: 1) relationships between submatrix elements of binary operators and coefficients of fractional parentage; 2) five conditions of "orthogonality" of coefficients of fractional parentage; 3) four summation conventions for submatrix elements of Tk, operator representing the sum of unity operators tk of k rank that are unitary in the j-space; 4) relationships between submatrix elements and coefficients of fractional parentage. The results obtained in this article link the matrix elements of the paired operators A+, B, quasi-spin operators, creation and annihilation operators of the electron, and coefficients of fractional parentage. In the future, they can significantly simplify studies related to calculating the matrix elements of the relativistic Hamiltonian of many-electron atoms and ions.