WikiMechanics

Consider a particle P, that is characterized by the coefficients of its rotating seeds $\; \it{N}^{\, \mathsf{U}}$ and $\it{N}^{\, \mathsf{D}}$. We say that P has a spin that depends on these coefficients, as noted in the accompanying table. We also define a number $\delta_{z}$, called the helicity, by

$\delta _{z} \equiv \begin{cases} +1 \; \; &\sf{\text{if}} \; \; &N^{\mathsf{U}} \; > \; N^{\mathsf{D}} \\ \; \; 0 \; \; &\sf{\text{if}} \; \; &N^{\mathsf{U}} \; = \, N^{\mathsf{D}} \\ -1 \; \; &\sf{\text{if}} \; \; &N^{\mathsf{U}} \; < \; N^{\mathsf{D}} \end{cases}$

which is used for making quantitative descriptions of P's spatial orientation. And later, if P is also being used as a frame of reference, then $\; \it{N}^{\, \mathsf{U}}$ and $\it{N}^{\, \mathsf{D}}$ may be used to establish the phase of other particles. So rotating seeds have primary role in describing motion. This task is expanded by considering the coefficients of leptonic seeds; $\, \it{N}^{\, \mathsf{A}}$, $\, \it{N}^{\, \mathsf{M}}$, $\, \it{N}^{\, \mathsf{G}}$ and $\, \it{N}^{\, \mathsf{E}}$, to define

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

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

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

where $\delta _{\hat{e}}$ is the electric polarity and $\delta _{\hat{m}}$ is the magnetic polarity. Then we specify the total angular momentum vector as $\begin{align} {\rm{\overline{J}}} \equiv \left( {\rm{J}}_{m} , \ {\rm{J}}_{e} , \ {\rm{J}}_{z} \right) \end{align}$. Exchanging quarks for anti-quarks does not alter seed counts, so ${\rm{\overline{J}}} ( \sf{P} ) = {\rm{\overline{J}}} ( \sf{\overline{P}} )$. 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 a solitary, undivided particle that is not electrically or magnetically polarized we may construct a framework where P is centered on the electric and magnetic axes. Then it is easy to assess the norm of ${\rm{\overline{J}}}$ because $N^{\mathsf{A}} = N^{\mathsf{M}}$ and $N^{\mathsf{G}} = N^{\mathsf{E}}$. The vector is aligned with the polar-axis, $\, {\rm{\overline{J}}} = \left( 0, \ 0, \ {\rm{J}}_{z} \right)$ and

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

This expression is simplified by defining the total angular momentum quantum number as

If $𝘑 \ne 0$ then the $z$-component of the angular momentum vector can be expressed in terms of $𝘑$ as

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

And if $𝘑 \gg 1$ then the radical is approximately one, and

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

Similar results obtain for the other axes so that

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

associative properties

Sensory interpretation: Rotating seeds are objectified from achromatic visual sensations, so for spin-up particles, white sensations outnumber black sensations. Collectively they are bright. For spin-down particles, black sensations are more numerous than white sensations, they look dark. Non-rotating particles are objectified from colourless sensations. So $\delta_{z}$ indicates if a complex achromatic visual sensation is brighter or darker than some medium grey visual experience. And $𝘑$ notes the size of the difference.

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Angular Momentum