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

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

And let each quark be described by its phase $\delta _{\theta}$ along with its radius vectpr $\overline{\rho}$. Definition: the **wavevector** of P is

$\begin{align} \overline{\kappa} \equiv \left( \frac{1}{\rho_{\LARGE{\bullet}}^{2}} - \frac{1}{\rho_{\LARGE{\circ}}^{2}} \right) \sum_{i=1} ^{N} \delta _{\theta}^{\, i} \; \overline{\rho}^{i} \end{align}$

where $\rho_{\LARGE{\bullet}}$ is the inner radius and $\rho_{\LARGE{\circ}}$ is the outer radius of P. Substituting-in the definitions for these radii gives the wavevector in terms of quark coefficients as

$\begin{align} \overline{\kappa} = \frac{ k_{\sf{F}}}{hc} \left[ \left( \frac{8}{ \Delta n^{\sf{D}} } \right)^{2} - \left( \frac{8}{ N^{\sf{D}} } \right)^{2} \right] \cdot \sum_{i=1} ^{N} \delta _{\theta}^{\, i} \; \overline{\rho}^{i} \end{align}$

But a free particle has $\left| \Delta n^{\sf{D}} \right| = 8$ and $N^{\sf{D}} \! \to ∞$. So the wavevector of a free particle can be put plainly as

$\begin{align} \overline{\kappa} = \frac{ k_{\sf{F}} }{ hc} \sum_{i=1} ^{N} \delta _{\theta}^{\, i} \; \overline{\rho}^{i} \end{align}$

Note that $\overline{\kappa}$ is a relative characteristic because the phase depends on the juxtaposition of P with some frame of reference. Let this frame be steady so that quarks do not change phase. Then if some free particles interact like $\mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z}$ their wavevectors will combine as

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

because quarks are indestructible and $\overline{\rho}$ is defined from sums and differences of quark coefficients. The radius vector also relates particles and anti-particles as $\, \overline{\rho} {\sf{(P)}} = - \overline{\rho} \sf{(\overline{P})}$. So if quarks are swapped with anti-quarks without altering the phase, then

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

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

$\lambda \equiv \begin{cases} \ \ \ 0 \ &\sf{\text{if}} \ &\kappa =0 \\ \ 1 / \, \kappa \ &\sf{\text{if}} \ &\kappa \ne 0 \end{cases}$