particles are connected through the relativistic dispersion relation established by Paul Dirac: which in the ultrarelativistic limit is and in the nonrelativistic limit is where is the invariant mass. In the nonrelativistic limit, is a constant, and is the familiar kinetic energy expressed in terms of the momentum . The transition from ultrarelativistic to nonrelativistic behaviour shows up as a slope change from p to p2 as shown in the log–log dispersion plot of E vs. p. Elementary particles, atomic nuclei, atoms, and even molecules behave in some contexts as matter waves. According to the de Broglie relations, their kinetic energy E can be expressed as a frequency ω, and their momentum p as a wavenumber k, using the reduced Planck constant ħ: Accordingly, angular frequency and wavenumber are connected through a dispersion relation, which in the nonrelativistic limit reads {| class="toccolours collapsible collapsed" width="60%" style="text-align:left" ! Animation: phase and group velocity of electrons |- | This animation portrays the de Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4 ångströms in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed, so that its group velocity is 0.707 c. The top electron has twice the momentum, while the bottom electron has half. Note that as the momentum increases, the phase velocity decreases down to c, whereas the group velocity increases up to c, until the wave packet and its phase maxima move together near the speed of light, whereas the wavelength continues to decrease without bound. Both transverse and longitudinal coherence widths (packet sizes) of such high energy electrons in the lab may be orders of magnitude larger than the ones shown here. |} Frequency versus wavenumber As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the refractive index—it is common to refer to the functional dependence of angular frequency on wavenumber as the dispersion relation. For particles, this translates to a knowledge of energy as a function of momentum. Waves and optics The name "dispersion relation" originally comes from optics. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant index of refraction, or by using light in a non-uniform medium such as a waveguide. In this case, the waveform