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

$\begin{align} \overline{p} = \frac{ h }{ 2 \pi } \overline{ \kappa }^{ \sf{P}} \end{align}$

Recall that for particles in motion the wavelength is given by

$\begin{align} \lambda = \frac{2 \pi }{ \kappa } \end{align}$

So taking the norm of the momentum and eliminating the wavenumber obtains Louis de Broglie'sXlink.png statement about the relationship between momentum and wavelength

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

DeBroglie's postulate notes a proportionality between $\overline{p}$ and $\overline{ \kappa }$ that is simply built into the definitions used by WikiMechanics.

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