MagicNumbers

If you're looking to calculate magic numbers from first principles, that's a pretty ambitious but exciting endeavor. The complexity arises from the many-body nature of the problem, as well as from the specific types of interactions among nucleons. Here are some steps and equations to consider:

Schrödinger Equation for Nucleon

You'll start with the non-relativistic Schrödinger equation for a nucleon (either a proton or a neutron):

$$ H \psi = E \psi $$

The Hamiltonian $H$ can be expressed as:

$$ H = T + V $$

where $T$ is the kinetic energy operator and $V$ is the potential energy operator.

Kinetic Energy Operator

The kinetic energy operator $T$ for a single nucleon is:

$$ T = -\frac{\hbar^2}{2m} \nabla^2 $$

Potential Energy Operator

The potential energy operator $V$ is a more complex term, usually modeled using the Woods-Saxon potential or a similar form, and is dependent on both radial distance $r$ and angular momentum $\ell$:

$$ V(r) = V_0 + V_{\text{SO}} \vec{L} \cdot \vec{S} $$

Here $V_0$ is the main potential, $V_{\text{SO}}$ is the spin-orbit coupling term, $\vec{L}$ is the angular momentum, and $\vec{S}$ is the spin of the nucleon.

Many-Body Problem

The Schrödinger equation needs to be generalized to a many-body equation for $A$ nucleons:

$$ H \Psi = E \Psi $$

where $\Psi$ is now a many-body wavefunction, and $H$ includes terms for every nucleon's kinetic and potential energy, as well as interactions between every pair of nucleons:

$$ H = \sum_{i=1}^{A} T_i + \sum_{i=1}^{A} V_i + \sum_{i < j} V_{ij} $$

Methods to Tackle the Many-Body Problem

1. Mean Field Approximation: One common approach is to use a mean-field approximation, where each nucleon moves in an average potential generated by all the other nucleons. The Hartree-Fock method is often used for this.

2. Configuration Interaction (CI) Methods: These are numerical methods to find an approximate solution to the many-body Schrödinger equation. They work by expressing the many-body wave function as a linear combination of a basis of Slater determinants.

3. Monte Carlo Methods: For very heavy nuclei, Monte Carlo methods like Green's Function Monte Carlo (GFMC) can be used.

4. Density Functional Theory (DFT): Sometimes used to describe the behavior of the nucleons in terms of a functional of the density.

5. Effective Field Theories: These theories aim to describe the essential physics of the problem by focusing on the most relevant degrees of freedom and interactions.

Remember, to solve for magic numbers, you're interested in situations where the shells are fully occupied, leading to higher stability.

I hope this gives you a starting point. Given the complexity, computational tools will likely be necessary unless you have some new insights to simplify the problem.