Orbital Angular Momentum

Definition: The spin quantum number $s$

$\begin{align} s \ = \ \frac{N^{\sf{U}} + N^{\sf{D}}}{8} -2{\mathrm{n}} \end{align}$

$\begin{align} s \ = \ \frac{N^{\sf{U}} - 3n^{\sf{d}} + n^{\sf{\bar{d}}} }{8} \end{align}$


Definition: The azimuthal quantum number $\ell$

$\begin{align} \ell \ = \ \frac{ \left( N^{\sf{U}} + N^{\sf{D}} \right) + \left| N^{\sf{U}} - N^{\sf{D}} \right| - 4n^{\sf{d}} }{8} \end{align}$

$\begin{align} \ell \ = \ \frac{ \left( N^{\sf{U}} + N^{\sf{D}} \right) + \left| N^{\sf{U}} - N^{\sf{D}} \right| - 16{\mathrm{n}} }{8} \end{align}$

$\begin{align} \ell \ = \ 𝘑 + \frac{ N^{\sf{U}} + N^{\sf{D}} -16{\mathrm{n}} }{8} \end{align}$

$\begin{align} \ell \ = \ 𝘑 - 2{\mathrm{n}} + \frac{ N^{\sf{U}} + N^{\sf{D}} }{8} \end{align}$


Definition: total angular momentum quantum number for excited atoms $j$

$\begin{align} j = \left| \, \ell - s \, \right| \end{align}$

$\begin{align} j = \left| \, 𝘑 - 2{\mathrm{n}} + \frac{ N^{\sf{U}} + N^{\sf{D}} }{8} - s \, \right| \end{align}$

$\begin{align} j = \left| \, 𝘑 - 2{\mathrm{n}} + \frac{ N^{\sf{U}} + N^{\sf{D}} }{8} - \left( \frac{N^{\sf{U}} + N^{\sf{D}}}{8} -2{\mathrm{n}} \right) \, \right| \end{align}$

$j = 𝘑$

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