Field Theoretic Approach to Atomic Helium

Abstract

Publisher Summary The simplest few-body problem is the hydrogen atom for which the Green's function method gives the Schrodinger equation, and is thus trivial. The helium isoelectronic series is the nontrivial few-body problem, which is discussed in this chapter. The investigation illustrates the application of rather formal field theoretic methods and diagrams to a simple system, which is familiar. The singlet and triplet states behave like boson and fermion systems respectively, since the spin will be separated out from the beginning. The Green's functions are taken with respect to the ground state of a one-particle system, and the poles thus occur for a two particle and a no-particle system. The chapter deals with the application of the many-body Green's function method to atomic hydrogen in order to illustrate the technique. The Hamiltonian and state vectors for the helium atom are written in second quantization. The Green's function and its spectral representation are introduced. The single-particle Green's function equation of motion is obtained. Further, the two-particle Green's function for the triplet state is discussed. The perturbation expansion for the single-particle Green's function is obtained in for the triplet state. The factorization of the two-particle Green's function is investigated. A comparison of the Green's function method with Hartree-Fock theory is drawn. The two-particle Green's function is factorized for the singlet state. The approximate equation for the single-particle propagator is obtained for the singlet state.