Orbital Angular Momentum

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Notice: this page is under construction

Definition: The spin angular momentum quantum number $s$
Definition: The azimuthal quantum number $\ell$
Definition: The total atomic angular momentum quantum number $j$

Quantum Numbers for Atoms
principal $\begin{align} {\mathrm{n}} \equiv \frac{ n^{\mathsf{d}} }{4} \end{align}$
azimuthal $\begin{align} \ell \equiv \frac{ N^{\sf{U}} + N^{\sf{D}} + \left| N^{\sf{U}} - N^{\sf{D}} \right| - 4{ n^{\sf{d}}} }{8} \end{align}$
spin angular momentum $\begin{align} s \equiv \frac{ n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} -3 n^{\mathsf{d}}+ n^{\mathsf{\overline{d}}} }{ 8 } \end{align}$
total atomic angular momentum $\begin{align} j \equiv \frac{ \, \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| \, }{8} \end{align}$

Relations between Quantum Numbers

$j = \ell - s$

$j = 𝘑$

$N^{\sf{U}} + N^{\sf{D}} = 8s + 16{\mathrm{n}}$

$N^{\sf{U}} - N^{\sf{D}} = 8 j \delta _{z}$


  • up date symbol for m /ell
  • discuss j vs J
  • define associated vectors

Definition: the quantum of orbital angular momentum, 𝜴. The union of a spin-up field with a spin-down field is called a quantum of orbital angular momentum.

${\mathit{\Omega}} \equiv \left\{ \, {\mathscr{F}} \! \! \uparrow, \ {\mathscr{F}} \! \! \downarrow \vphantom{H^H} \right\}$

so that

${\mathit{\Omega}} \leftrightarrow 2 {\sf{u}} + 2 {\sf{\overline{u}}} + 4 {\sf{\overline{d}}}$

Absorbing or emitting ${\mathit{\Omega}}$ changes the azimuthal quantum number by ±1 without altering the principal quantum number, or the total angular momentum quantum number.

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