1) Obtain 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 appropriate frecuency), the charge density of 16O, and the elastic dispersion cross section of 400 MeV electrons.

This question is beyond me. I have not looked at this material for years. Hopefully someone who is up to date will see it.

These are major undertakings worthy of a PhD thesis when I went to grad school. Is this a single night's homework assignment? I find that hard to believe.

I don't know why you bother to give the atomic weight of the atoms, since only the atomic number affects the wave functions.

I suspect this is a made-up prank question, or a PhD thesis assigment. Wave functions do not have frequencies. Charge densities can be calculated fairly well with the Thomas-Fermi model, described at

(Broken Link Removed)

I recommend using the Born-Oppenheimer approximation for the elastic scattering cross section of high energy electrons. See
http://www.springerlink.com/content/w40m51740685764j/fulltext.pdf?page=1
for details.

Good luck with your thesis.

To obtain the binding energies of 4He and 18O in the Hartree-Fock approximation, we need to follow these steps:

1) Set up the Hartree-Fock equations for the nuclei.

2) Solve the Hartree-Fock equations numerically using a variational approach.

3) Calculate the total binding energy from the Hartree-Fock wave function.

4) Repeat the process for both 4He and 18O.

Here's a step-by-step explanation for each step:

1) Set up the Hartree-Fock equations:
The Hartree-Fock equations describe the self-consistent field approximation for the many-body system. In this case, we are interested in the nuclei of 4He and 18O. The equations can be written as follows:

(-ħ^2/2μ)∇^2ψ + UHFψ = εψ

Where (-ħ^2/2μ) is the kinetic energy operator, ψ is the wave function, UHF is the Hartree-Fock potential, ε is the energy eigenvalue, and ∇^2 is the Laplacian operator.

2) Solve the Hartree-Fock equations numerically:
To solve the Hartree-Fock equations, we need to start with an initial set of guess wave functions, and then iteratively solve for the self-consistent field. This requires diagonalizing the Hamiltonian matrix until convergence is achieved. The iterative process is usually performed using computer programs specifically designed for this purpose, such as the HFODD (Hartree-Fock-Bogoliubov (HFB) Solver Using the Gogny Interaction) code.

3) Calculate the total binding energy:
Once the self-consistent field is obtained, we can calculate the total binding energy by summing up the individual energies of the occupied orbitals. The binding energy is given by the sum of the eigenvalues of the occupied orbitals, multiplied by their respective occupation numbers.

4) Repeat for 4He and 18O:
Repeat the above steps for both 4He and 18O nuclei, using the appropriate input parameters for each system.

Now, moving on to the second question:

To calculate the charge density of 16O using oscillator wave functions with the appropriate frequency:

1) Choose an appropriate oscillator basis:
In this case, we need to choose an oscillator basis that is well-suited for describing the charge density distribution of 16O. The basis functions are typically harmonic oscillator wave functions.

2) Calculate the wave function for the 16O nucleus:
Using the chosen oscillator basis, construct the wave function for the 16O nucleus. This can be done by diagonalizing the appropriate Hamiltonian matrix and solving for the eigenfunctions and eigenvalues.

3) Calculate the charge density:
The charge density can be obtained by squaring the wave function obtained in step 2. The charge density represents the spatial distribution of the positive charge in the nucleus.

Now, for the third part of the question, calculating the elastic dispersion cross section of 400 MeV electrons:

1) Define the problem:
The elastic dispersion cross section describes the scattering of electrons off a target nucleus. In this case, we are interested in the dispersion of 400 MeV electrons off an unknown target nucleus.

2) Use appropriate scattering theory:
To calculate the elastic dispersion cross section, one needs to employ scattering theory. This involves solving the scattering equations or using scattering approximations, such as the Born approximation or the eikonal approximation.

3) Input appropriate parameters:
To perform the scattering calculation, we need to input the appropriate parameters such as the energy of the electrons (in this case, 400 MeV), the charge density of the target nucleus, and the scattering potential.

4) Calculate the cross section:
Using the chosen scattering theory and the input parameters, calculate the elastic dispersion cross section. The cross section represents the probability of scattering of the electrons off the target nucleus at a specific angle.

It's important to note that both calculations (charge density and elastic dispersion cross section) involve advanced theoretical methods and often require the use of specialized computational techniques and software.