This section we will be solving Schrodinger’s equation, numerically as analytical solutions are too complicated, for a suitable potential to look into nuclei
If we take a plot of the SEMF prediction - the actual binding energy we get:
We se a grid like pattern known as nuclear magic numbers and for when $N$ and $Z$ are “magical” it is called doubly magic nuclei
Experimentally these numbers were found to be
$$ \begin{aligned} N&=2,8,20,28,50,82,126 \\ Z&=2,8,20,28,50,82
\end{aligned} $$
🗒️ Note: we will use a mean field approximation whereby all interactions of one nucleon with all other averages out to a simple potential that depends on the nucleon’s location
<aside> <img src="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/46902408-b9b5-4f8d-9e27-d3ed3490f157/Mean_field_approximation.gif" alt="https://prod-files-secure.s3.us-west-2.amazonaws.com/369dfa6b-d4d9-4cf2-a446-e369553b6347/46902408-b9b5-4f8d-9e27-d3ed3490f157/Mean_field_approximation.gif" width="40px" />
Mean field approximation: all interactions of one nucleon with all other averages out to a simple potential that depends on the nucleon’s location
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🗒️ Note: we use this approximation to make the Schrodinger equation solvable
$$ -\frac{\hbar^2}{2M} \nabla^2 \psi(\vec r)+\frac 12 M\omega ^2 r^2 \psi(\vec r)=E\psi(\vec r) $$
where $M$ and $\psi$ refer to single nucleon mass and wave function
Splitting the equation into 3 $\rm 1D$ equations the eigenvalues are
$$ E_{n_xn_yn_z}=(n_x+n_y+n_z+3/2)\hbar \omega $$
Since degeneracy is key we will write $\psi(\vec r)=R_{nl}(r)Y_{lm}(\theta,\phi)$
$$ -\frac{\hbar^2}{2M}\frac{1}{r^2}\frac{\text d}{\text dr} \left ( r^2 \frac{\text d}{\text dr} R(r) \right )+\frac{\hbar^2 l(l+1)}{r^2} R(r)+\frac 12 M\omega^2 r^2 R(r)=E_{nl}R(r) $$
which gives us the energies as
$$ E_{nl}=(2n+l+3/2)\hbar\omega $$
where $n\in \N$ and $l\in \N$ is the angular momentum of the state
🗒️ Note: $l=0,1,2,3,\ldots$ corresponds to $s,p,d,f,\ldots$
🧠 Remember:
🗒️ Notes: no restriction on $l$ by $n$ as there is no potential proportional to $1/r$
If we look at the shells
