Total Angular Momentum
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Notice: this page is under construction

Particle Type Definition
a spin-up particle $N^{ \mathsf{U}} > N^{ \mathsf{D}}$
a non-rotating particle $N^{ \mathsf{U}} = N^{ \mathsf{D}}$
a spin-down particle $N^{ \mathsf{U}} < N^{ \mathsf{D}}$

Consider a particle P that is characterized by the coefficients of its rotating seeds, $\it{N}^{\, \mathsf{U}}$ and $\it{N}^{\, \mathsf{D}}$. Particles are classified by their rotating seed content as noted in the accompanying table.

Related to the $\delta_{z}$, the helicity, and also the phase.
And also solar clocks.


Definition of Angular Momentum Vector

$\begin{align} {\rm{J}}_{z} \equiv \delta_{z} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, \left| N^{\mathsf{U}}-N^{\mathsf{D}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}$

$\begin{align} {\rm{J}}_{m} \equiv \delta _{\hat{m}} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{A}}-N^{\mathsf{M}} \right)^{2} + 8 \, \left| N^{\mathsf{A}}-N^{\mathsf{M}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}$

$\begin{align} {\rm{J}}_{e} \equiv \delta _{\hat{e}} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{G}}-N^{\mathsf{E}} \right)^{2} + 8 \, \left| N^{\mathsf{G}}-N^{\mathsf{E}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}$

where

$\delta _{\hat{e}}$ is the electric polarity

$\delta _{\hat{m}}$ is the magnetic polarity

the total angular momentum vector

$\begin{align} {\rm{\overline{J}}} \equiv \left( {\rm{J}}_{m} , \ {\rm{J}}_{e} , \ {\rm{J}}_{z} \right) \end{align}$


Definition of Total Angular Momentum Quantum Number

In general, the components ${\rm{J}}_{m}$, ${\rm{J}}_{e}$ and ${\rm{J}}_{z}$ have non-zero values, and a P's motion is complicated.

But for an solitary, undivided, whole particle that is not electrically or magnetically polarized we can have a frame-of-reference where P is centered on the electric and magnetic axes.

In this particle-centered frame, it easy to assess the norm of ${\rm{\overline{J}}}$ because $N^{\mathsf{A}} = N^{\mathsf{M}}$ and $N^{\mathsf{G}} = N^{\mathsf{E}}$. Then ${\rm{\overline{J}}} = \left( 0, \ 0, \ {\rm{J}}_{z} \right)$ and

$\begin{align} {\large{\parallel}} \, {\rm{\overline{J}}} \, {\large{\parallel}} &= \left| \, {\rm{J}}_{z} \right| = \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, \left| N^{\mathsf{U}}-N^{\mathsf{D}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}$

The total angular momentum quantum number is defined as

$\begin{align} 𝘑 \equiv \frac{ \, \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| \, }{8} \end{align}$

then

$\begin{align} {\large{\parallel}} \, {\rm{\overline{J}}} \, {\large{\parallel}} = \frac{h}{\rm{2} \pi} \sqrt{ \, 𝘑 \, \left( 𝘑 + \rm{ 1} \right) \; \vphantom{1^{2}} } \end{align}$


Conservation of Angular Momentum

The $z$-component of the angular momentum vector can be expressed in terms of the angular momentum quantum number as

$\begin{align} {\rm{J}}_{z} = \delta_{z} \frac{h}{\rm{2} \pi} \sqrt{ \, 𝘑 \, \left( 𝘑 + \rm{ 1} \right) \; \vphantom{1^{2}} } \end{align}$

If $𝘑 \ne 0$ then

$\begin{align} {\rm{J}}_{z} = \delta_{z} \frac{𝘑 h}{\rm{2} \pi} \sqrt{ \, 1 + \frac{1}{ 𝘑 } \vphantom{{\frac{1}{ 𝘑 }}^9} \; } \end{align}$

If $𝘑 \gg 1$ then

$\begin{align} {\rm{J}}_{z} &\simeq \delta_{z} \frac{𝘑 h}{\rm{2} \pi} \\ \\ &= \frac{h}{16 \pi} \delta_{z} \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| \\ \\ &= \frac{h }{16 \pi} \left( N^{\mathsf{U}} - N^{\mathsf{D}} \right) \end{align}$

And similarly for the other axes. Then

$\begin{align} \overline{{\rm{J}}} \equiv \left( {\rm{J}}_{m} , \ {\rm{J}}_{e} , \ {\rm{J}}_{z} \right) \simeq \frac{h }{16 \pi} \left( N^{\mathsf{A}} - N^{\mathsf{M}}, \ \ N^{\mathsf{G}} - N^{\mathsf{E}}, \ \ N^{\mathsf{U}} - N^{\mathsf{D}} \vphantom{W^{W^{W}}} \right) \end{align}$

Seeds are indestructible. All conservation laws are logically based on the conservation of seeds. They are always conserved when particles interact. Therefore the total angular momentum vector is always approximately conserved too. For macroscopic particles, $𝘑$ is huge because $h$ is so small, and the approximation is excellent.


Exchanging quarks for anti-quarks does not alter thermodynamic seed counts, so for the angular momentum

${\rm{\overline{J}}} \left( \sf{P} \right) = {\rm{\overline{J}}} \left( \sf{\overline{P}} \right)$


//Conus,// Jean Baptiste Lamarck. Tableau Encyclopédique et Méthodique des Trois Regnes de la Nature, Paris 1791-1798. Photograph by D Dunlop.
Conus, Jean Baptiste Lamarck. Tableau Encyclopédique et Méthodique des Trois Regnes de la Nature, Paris 1791-1798. Photograph by D Dunlop.
achromatic.jpg
Sensory interpretation: Rotating seeds are objectified from achromatic visual sensations, so for spin-up particles, white sensations outnumber black sensations. Collectively they are bright sensations. For spin-down particles, black sensations are more numerous than white sensations, they look dark. So spin can be understood as a description of brightness. Non-rotating particles are objectified from sensations that are mostly dull or grey.

Summary
Adjective Definition
Total Angular Momentum Quantum Number $\begin{align} 𝘑 \equiv \frac{ \, {\large{\mid}} \, N^{\mathsf{U}} \! -N^{\mathsf{D}} {\large{\mid}} \, }{8} \end{align}$ 4-11
Noun Definition
Spin-Up Particles $N^{ \mathsf{U}} > N^{ \mathsf{D}}$ 5-25
Noun Definition
Spin-Down Particles $N^{ \mathsf{U}} < N^{ \mathsf{D}}$ 5-27
Noun Definition
Non-Rotating Particles $N^{ \mathsf{U}} = N^{ \mathsf{D}}$ 5-26
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