Spin Angular Momentum
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 $N^{ \mathsf{U}} < N^{ \mathsf{D}}$
//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.

Consider a particle P characterized by $\it{N}^{\, \mathsf{U}}$ and $\it{N}^{\, \mathsf{D}}$ the coefficients of its rotating seeds. Definition: The spin angular momentum of P is

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

Recall that the spin quantum number of P is defined as

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

So $\it{\Sigma}$ can be written in terms of $\sigma$ as

$\begin{align} \it{\Sigma} = \frac{h}{\rm{2} \pi} \sqrt{ \, \sigma \, \left( \sigma + \rm{1} \right) \; \vphantom{1^{2}} } \end{align}$

Particles are classified by their rotating seed content as noted in the accompanying table.

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 overall dull or grey.

Summary
Noun Definition
Spin-Up Particles $N^{ \mathsf{U}} > N^{ \mathsf{D}}$ 5-21
Noun Definition
Spin-Down Particles $N^{ \mathsf{U}} < N^{ \mathsf{D}}$ 5-23
Noun Definition
Non-Rotating Particles $N^{ \mathsf{U}} = N^{ \mathsf{D}}$ 5-22
Adjective Definition
Spin Angular Momentum $\begin{align} {\it{\Sigma}} \equiv \frac{h}{16 \pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} +8 \mid \, N^{\mathsf{U}}-N^{\mathsf{D}} \mid \, \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}$ 5-24
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License