Tampan, Paminggir people. Sumatra 19th century, 58 x 61 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop. |

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 $\tilde{ \kappa }$ the average wavevector of the quarks in F. The momentum of P in the F-frame is defined as

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

Let the frame of reference be formed from a gravitational component $\mathbb{G}$ and a smaller part $\mathbb{S}$ that surrounds P so that

$\overline{ \kappa }^{ \sf{F}} = \overline{ \kappa }^{ \mathbb{G}} + \overline{ \kappa }^{ \mathbb{S}}$

When gravitational effects are completely negligible

$\overline{\kappa} ^ { \mathbb{G} } = (0, 0, 0)$

and then P's momentum is given by

$\begin{align} \overline{p} = h \left( \overline{ \kappa }^{ \sf{P}} - \frac{ N^{ \sf{P}} }{ N^{\sf{F}} } \overline{ \kappa }^{ \mathbb{S}} \right) \end{align}$

From de Broglie's postulate we have $\, \lambda = h / p \,$, so

$\begin{align} \lambda = \frac{1}{ \left\| \, \overline{ \kappa }^{ \sf{P}} - \frac{ N^{ \sf{P}} }{ N^{\sf{F}} } \overline{ \kappa }^{ \mathbb{S}} \right\| } \end{align}$

To simplify, set $\overline{\kappa} ^ { \mathbb{S} } = (0, 0, 0)$ in the expression above to define $\lambda_{\sf{o}}$ as the wavelength of P in a non-dispersive medium

$\begin{align} \lambda_{\sf{o}} \equiv \frac{1}{ \left\| \, \overline{ \kappa }^{ \sf{P}} \right\| } \end{align}$

Many different combinations of photons and media are usefully characterized by the ratio

$\begin{align} \eta \equiv \frac { \, \lambda_{ \sf{o}} }{ \lambda } \end{align}$

Then in environments where there is no dispersion, $\lambda = \lambda_{ \sf{o}}$ and motion is described by $\eta =1$. Definition: the number $\eta$ is called the **index of refraction**.