Phase Angle
//Melethallia (detail),// Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.
Melethallia (detail), Ernst Haeckel, Kunstformen der Natur. Chromolithograph 32 x 40 cm, Verlag des Bibliographischen Instituts, Leipzig 1899-1904. Photograph by D Dunlop.

Let particle P be described by an ordered chain of events

$\Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \ \ldots \ \sf{P}_{\it{k}} \ \ldots \ \right)$

that is repetitive so that $\Psi$ may also be written as

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2} , \sf{\Omega}_{3} \ \ldots \ \right)$

where each orbital cycle $\sf{\Omega}$ is composed of $N$ sub-orbital events

$\sf{\Omega} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \ \ldots \ \sf{P}_{\it{N}} \right)$

If P contains just a few quarks then the time of occurrence may not be a useful parameter for describing these events because Planck's postulate is plausibly justified on a statistical basis. So we also discuss the order of events using a phase angle defined by

$\begin{align} \theta_{ k} \equiv \theta_{\sf{0}} - \delta_{z}\frac{2\pi k}{ N } \end{align}$

where $\theta_{0}$ is arbitrary and $\delta_{z}$ is the helicity of P. The ground-states and most excited-states of atoms have spin-down orientations, then $\delta_{z}=-1$. Also, the change in $\theta$ during one sub-orbital event is called the phase angle increment. It is written as $d\theta \equiv 2 \pi/ N$, so usually

$\begin{align} \theta_{ k} = \theta_{\sf{0}} +k \hspace{2px} d\theta \end{align}$

WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events $N$ may be large but not infinite. This requirement can be relaxed later to make a continuous approximation, thereby allowing the use of calculus. But in principle $N$ is finite and accordingly changes in $\theta$ may be small but not infinitesimal. For isolated particles the increment in the phase angle does not vary and so there is an equipartition of $\theta$ between sub-orbital events regardless of their quark content.

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