The methods discussed have been developed with the nuclear many-body problem, formulated as a classical quantum mechanical many-body problem with only twoparticle interaction and without any assumption of the origin of nuclear forces in mind. Two approaches characterized by different wave functions are discussed, and the general equations for the determination of best twoparticle orbitals'' are derived. The problem of the two interacting particles are examined in a common potential. This common potential is chosen as an oscillator. (W.D.M.)
The methods discussed have been developed with the nuclear many-body problem, formulated as a classical quantum mechanical many-body problem with only twoparticle interaction and without any assumption of the origin of nuclear forces in mind. Two approaches characterized by different wave functions are discussed, and the general equations for the determination of best twoparticle orbitals'' are derived. The problem of the two interacting particles are examined in a common potential. This common potential is chosen as an oscillator. (W.D.M.)
A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of N fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and f(2)(ij) is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained N one-particle equations for the N one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.
A theory is formulated to derive an effective interaction in nuclei in the framework of the unitarymodel-operator approach.A unitary operator is introduced to describe two-body correlations, and it is determined from the condition that the effective interaction should be decoupled between the lowmomentum model space and the high-momentum space.The unitary-transformed Hamiltonian is represented in a cluster expansion form.The one-and two-body cluster terms are mainly discussed.The theory is applied to the calculation of the energies of the ground-and one-body states with the Reid soft core potential.A large gain of the energy has been obtained for the ground state in comparison with the result of the usual G-matrix theory.As for the one-body energies obtained, agreement with the experimental values is fairly good.Although the discrepancies with the experiments still remain at the 1O~ 15 % level, the present approach shows to have some desirable features compared with the nuclear many-body theories proposed so far. § 1. IntroductionA fundamental problem in nuclear physics is to clarify the properties of nuclei in terms of the interaction among their constituent particles.Although there have been many theories or approaches, we have not yet reached the goal of reproducing successfully the experimental data.The investigation of nuclear properties, in the usual nonrelativistic approach, proceeds in two steps: The first one is to determine the interaction among nucleons, and the second is to solve the many-body SchrOdinger equation.Concerning the nucleon-nucleon force, it seems that the phenomenological potential, such as the Reid potential, would be incomplete and one should wait until quark dynamics, probably quantum chromodynamics (QCD), solves finally the problem o~ nuclear force.It is of no doubt that in order to fit the experiments we need more accurate knowledge on the nuclear force including the threebody force.However, if we want to reach the goal, we need also a good many-body theory.Regardless of the accuracy of the nuclear force, we shall not be able to solve the problem unless we have a way of constructing an accurate solution for a many-body system.We believe that many-body theory should share the responsibility for the discrepancy between theory and experiment.It will be sure that nuclear many-body theory is still to be improved.Among many theories for nuclear many-body problem, the most widely applied method is, needless to say, the linked-diagram perturbation theory.This approach may also be referred to as the folded-diagram theory based on the G-matrix interaction especially due 'to Brandow 1l and Kuo et al. 2l An outstanding problem in this approach is the convergence of the perturbation expansion.Since Schucan and Weidenmliller 3 ) pointed out the divergence of the perturbation expansion due to the presence of intruder states, this difficulty has long been a serious problem.Present status of this problem is
We extend a method previously presented for the three-body problem with attractive interparticle potentials to the case of repulsive interparticle potentials, with periodic boundary conditions on each particle. As before, we decompose the wave function into three parts, and from the Schrödinger equation write equations for these parts. We observe that, in one-dimension and for δ-function potentials,these equations can be easily solved numerically, and we present these solutions for potential“strengths” ranging from zero to infinity. We then discuss a more general method for solving these equations, which method involves expansions in a certain set of two-body functions. This general method is not necessarily limited to either one dimension or δ-function potentials, but, as a check on it, we do apply it to that case and get good agreement with the previous numerical results. As a further exploration of the method, we apply it to square well potentials in one dimension. In an Appendix we discuss the set of two-body functions that we use.
Abstract This chapter begins by defining the electronic structure problem of matter. Following the introduction of the system of atomic units, the electron-nuclear many-body problem is then formulated as a Schrödinger equation of the coupled electron-nuclear many-body system. The electronic and nuclear degrees of freedom are then decoupled using the Born-Oppenheimer approximation, which defines the electronic many-body problem. An illustration of potential-energy curves of a diatomic molecule is given, as well as an example of an interacting and a noninteracting two-electron system such as the helium atom. The chapter concludes by discussing the N-electron wave function, arguing that it becomes meaningless when the number of electrons, N, becomes large, due to the exponential increase of complexity.
For describing the pair correlations of the electrons in the many- electron systems a new method of reducing the many-particle problem to the two- particle problem is developed. Instead of the wave function, one- and twoparticle density matrices are used. In the process of deriving an equation for the two-particle wave function, which describes a pair of electrons in the field of the others, a transition to the second quantization representation is accomplished. The solution of the obtained equation is carried out with the help of perturbation theory. (auth)