Louis de Broglie's Postulate Louis de Broglie, 1892—1987.

Consider some particle P characterized by its wavevector $\overline{ \kappa }$ and the total number of quarks it contains $N$. Report on any changes relative to a frame of reference F which is characterized using the average wavevector$\ \ \tilde{ \kappa }$ of the quarks in F. The momentum of P in the F-frame is defined as

\begin{align} \overline{p} \equiv \frac{h}{2\pi} \left( \overline{ \kappa }^{ \sf{P}} \! - N^{ \sf{P}} \, \tilde{ \kappa }^{ \sf{F}} \right) \end{align}

But in a perfectly inertial frame of reference

$\tilde{ \kappa } ^{ \sf{F}}= \left( 0, 0, 0 \right)$

then the momentum of P is

\begin{align} \overline{p} = \frac{h}{2\pi} \overline{ \kappa } \end{align}

Recall that for particles in motion the wavelength is given by $\lambda = 2 \pi / \kappa$. So taking the norm of the momentum and eliminating the wavenumber obtains statement about the relationship between momentum and wavelength

\begin{align} \lambda = \frac{ h }{ p } \end{align}

DeBroglie's postulate notes a conditional proportionality between $\overline{p}$ and $\overline{ \kappa }$ that is simply built into the definitions used by WikiMechanics. Next step: slow motion.
page revision: 165, last edited: 14 Oct 2019 17:25