The Wavevector

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 and it has been contentious during the development of physics. However for WikiMechanics there is no quandary; scientific facts and theories are founded on sensation, and whether we call these perceptions particles or waves is just a question of convenience. If feelings are localized, then we talk about particles. Or if sensory phenomena seem to have some extended quality, then we often use words like wave, wavenumber, wavelength, etc. In between, we might speak of particles that are in excited states. These terms are developed from a discussion of quarks as follows.
 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.

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

$\lambda \equiv \begin{cases} \hspace{15 px} 0 \; &\sf{\text{if}} \; &\kappa =0 \\ \; 2 \pi / \kappa \; &\sf{\text{if}} \; &\kappa \ne 0 \end{cases}$

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.
 Next step: inertial frames.
page revision: 599, last edited: 05 Nov 2019 12:53