So far WikiMechanics has considered a generic particle P by objectifying some chain of events $\Psi$ written as
$\Psi ^{\sf{P}} = \left( \sf{\Omega} _{1} , \sf{\Omega} _{2} , \sf{\Omega} _{3} \ldots \right)$
The events $\sf{\Omega}$ are defined by sensation, and they are supposed to be very repetitive so that P can be recognized, $\; \sf{\Omega} _{1} = \sf{\Omega} _{2} = \sf{\Omega} _{3} \; \ldots$ etc. But we might just as well understand such a recurring sequence of sensations to be a wave train. That is, $\Psi$ could represent some sort of of periodically undulating or fluctuating perception. This interpretive ambivalence is called wave-particle duality
Let each orbital cycle of P be 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 for a perfectly free particle, $\left| \Delta n^{\sf{D}} \right| = 8$ 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
$\lambda \equiv \begin{cases} \hspace{15 px} 0 \; &\sf{\text{if}} \; &\kappa =0 \\ \; 2 \pi / \kappa \; &\sf{\text{if}} \; &\kappa \ne 0 \end{cases}$
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