Hartree fock calculation in nuclei

[bforcen]

Homework Statement

I need to solve these problems. Please help!:

1) Obtein the binding energies of 4He and 18O in the Hartree-Fock aproximmation. Use as variational space three oscillator shells for every occupied orbit. Don't use the spin-orbit coupling term.

2) Calculate (using oscillator wave functions with the appropiate frecuency), the charge density of 16O, and the elastic dispersion cross section of 400 MeV electrons.

Homework Equations

The Attempt at a Solution

 

[BigJDubs]

1) The Hartree-Fock approximation is an ansatz to solve the Schrödinger equation. That means that we have to solve a set of (coupled) equations. To do that, we need wave functions. In this case, we will use oscillator wave functions for every occupied orbit. Let's take 4He as example. It has two protons and two neutrons, so it has four fermions in total. We are using three oscillator shells for every occupied orbit, so we have nine shells in total. We denote the shells by n1, n2, n3, etc. The wave function of each fermion is a Slater determinant of single-particle wave functions. The single particle wave functions are given by:φ(n1)=Aexp(-βr2)where A is a normalization constant and β=mω/ℏ with m being the rest mass of the particle, ω the frequency of the oscillator and ℏ the reduced Planck's constant. We can then obtain the binding energy of 4He in Hartree-Fock approximation by solving the Hartree-Fock equation. This equation is of the form: E(n1,n2,n3...)=∑nΣiπ(ni)Ei + 1/2∑nΣi,jπ(ni,nj)Vij where Ei is the single particle energy of the i-th particle and Vij is the two-body interaction potential. Once we have solved the Hartree-Fock equation, we can obtain the binding energy of 4He. For 18O, we proceed in a similar way. 2) To calculate the charge density of 16O, we first need to calculate the wave function of the nucleus. This can be done using the oscillator wave functions with the appropiate frequency. Then, we can calculate the charge density from the wave function. To calculate the elastic dispersion cross section of 400 MeV electrons, we first need to calculate the scattering matrix. This can be done using the scattering theory. Once we have the scattering matrix, we can calculate the