Orbital Angular Momentum

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The direct discussion of visual sensation is curtailed as we objectify descriptions and apply the hypothesis of spatial isotropy. To accomplish this we have replaced the four coefficients of rotating quarks, that represent achromatic sensations, with new characteristics that are defined to specify the rotational state of an atom.

Atomic Quantum Numbers

Here are new quantum numbers for use in atomic descriptions that gloss over visual sensation. To discuss atoms objectively we adopt $\mathrm{n},$ $\ell,$ $s$ and $j$ instead of the four coefficients of rotating quarks $n^{\mathsf{u}},$ $n^{\mathsf{\overline{u}}},$ $n^{\mathsf{d}}$ and $n^{\mathsf{\overline{d}}}$ that previously accounted for achromatic sensation.

The principal quantum number is given by

$\mathrm{n} \equiv \dfrac{n^{\mathsf{d}}}{4}$

This principal number is the same quantity that was developed earlier for the description of excited particles.

The azimuthal quantum number is defined by

$\ell \equiv \dfrac{N^{\mathsf{U}} + N^{\mathsf{D}} + \left| N^{\mathsf{U}} - N^{\mathsf{D}} \rule{0px}{9px} \right| }{8} - \dfrac{n^{\mathsf{d}}}{2}$

The spin angular momentum quantum number is defined as

$s \equiv \dfrac{ n^{\mathsf{u}} + n^{\mathsf{\overline{u}}} - 3 n^{\mathsf{d}}+ n^{\mathsf{\overline{d}}} }{8}$

And finally, the total atomic angular momentum quantum number is

$j \equiv \dfrac{\left| N^{\mathsf{U}} - N^{\mathsf{D}} \rule{0px}{9px} \right|}{8}$

Some Useful Identities

$j = \ell - s$

$j = \textsl{\textsf{J}}$

$\left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| = \delta_{z} \left( N^{\mathsf{U}} - N^{\mathsf{D}} \right) = \delta_{z} N^{\mathsf{U}} - \delta_{z} N^{\mathsf{D}} = 8 \hspace{2px} j$

The Lamb Quantum

We use the letter $\text{\L}$ to note a quantum of orbital angular momentum. Absorbing or emitting $\text{\L}$ changes the azimuthal quantum number by $\Delta \ell = \pm 1$ without altering $\mathrm{n}$ or $j$ . This particle is used to explain the Lamb shift that is observed in the spectrum of hydrogen. So $\text{\L}$ is called the Lamb quantum and is given by

$\text{\L} \leftrightarrow 2 {\mathsf{u}} + 2 {\mathsf{\overline{u}}} + 4 {\mathsf{\overline{d}}} + 2 {\mathbf{l}} + 2 {\mathbf{\overline{l}}}$