The Wavevector
//Ajat basket//, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Hornbill motif. From the Teo Family collection, Kuching. Photograph by D Dunlop.
Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Hornbill motif. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Consider a particle P characterized by some repetitive chain of events

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

where each orbital cycle is a bundle of $N$ quarks

$\sf{\Omega}^{\sf{P}} = \left\{ \sf{q}^{\sf{1}}, \sf{q}^{\sf{2}} \ldots \sf{q}^{\it{i}} \ldots \sf{q}^{\it{N}} \right\}$

Let each quark be described by its phase $\delta _{\theta}$ and its radius vector $\overline{\rho}$. Definition: the wavevector of P is

$\begin{align} \overline{ \kappa }^{ \sf{P}} &\equiv \frac{1}{k_{\sf{A}}} \sum_{i=1} ^{N} \delta _{\theta}^{\, i} \; \bar{\rho}^{i} \end{align}$

where $k_{\sf{A}} \equiv hc \, / \, 2 \pi \, k_{\sf{F}}$ is a constant with the units of an area. So $\overline{\kappa}$ is a relative characteristic because the phase depends on some frame of reference. Let this frame be steady so that quark phases do not change, then if some compound quarks interact like $\mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z}$ their wavevectors will combine as

$\overline{\kappa}^{ \mathbb{X}} + \overline{\kappa}^{\mathbb{Y}} = \overline{\kappa}^{\mathbb{Z}}$

This holds because quarks are indestructible and $\overline{\rho}$ is defined from sums of quark coefficients. Theorem: particles and anti-particles have symmetrically opposed wavevectors

$\overline{\kappa} \left( \sf{P} \right) = - \, \overline{\kappa} \left( \overline{\sf{P}} \right)$

because $\overline{\rho} \left( \sf{P} \right) = - \it{ \overline{\rho}} \left( \sf{\overline{P}} \right)$. Definition: the average wavevector describes some hypothetical typical quark in P using the ratios $\tilde{\kappa} \equiv \overline{\kappa} / N$ where $N$ is the total number of all types of quarks in P. Definition: a wavenumber is the norm of a wavevector, and written without an overline as $\kappa \equiv \left\| \, \overline{\kappa} \, \right\|$. Definition: the wavelength of P is

$\lambda \equiv \begin{cases} \ \ \ 0 \ &\sf{\text{if}} \ &\kappa =0 \\ \ 2 \pi / \kappa \ &\sf{\text{if}} \ &\kappa \ne 0 \end{cases}$

achromatic.jpgBangladeshi.GIFSwedish.gifodd.jpg
Sensory Interpretation: Radius vectors depend on dynamic quarks, not baryonic quarks. So the wavevector, wavenumber and wavelength only represent somatic and visual sensations, not thermal perceptions.
Right.png Next step: inertial frames.
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