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Refraction

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

\overline{p} \equiv      \dfrac{h}{2\pi} \left(\overline{\kappa}^{\mathsf{P}} -  N^{\mathsf{P}} \, \widetilde{\kappa}^{\mathsf{F}} \right)

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

\mathsf{F} = \left\{ \mathbb{G}, \, \mathbb{S} \rule{0px}{10px} \right\}

and

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

Let the large part of F be responsible for any gravitons that interact with P so that when gravitational effects are completely negligible

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

Then P’s momentum is given by

\overline{p} = \dfrac{h}{2\pi} \left( \overline{\kappa}^{\mathsf{P}} - \dfrac{N^{\mathsf{P}}}{N^{\mathsf{F}}} \, \overline{\kappa}^{\mathbb{S}} \right)

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

\lambda = \dfrac{2\pi}{ \left\| \, \overline{\kappa}^{\mathsf{P}} - \dfrac{N^{\mathsf{P}}}{N^{\mathsf{F}}} \overline{\kappa}^{\mathbb{S}} \right\| }

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

\lambda_{\mathsf{o}} \equiv \dfrac{2\pi}{ \left\| \, \overline{\kappa}^{\mathsf{P}} \right\| }

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

\eta \equiv \dfrac{ \,  \lambda_{\mathsf{o}}}{\lambda}

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

Refraction describes the shimmering iridescence evoked by this lustrous silken textile from Indonesia.
Tampan, Paminggir people. Sumatra 19th century, 58 x 61 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.