Particle Properties of Radiant Energy in Wave Guides

The mean energy per excitation wavelength of e.m. radiation propagated along a lossless uniform wave guide is found to be $w = Qμ$⁠, $Q$ being independent of frequency for a given amplitude of the longitudinal field intensity, mode of the field, and structure of the guide. The frequency $μ$ and the group velocity $vg$ of the radiant energy are related by the equation $μ/μc = [1 − (vg/c)2]−12$ where $μc = c/λc$ denotes the frequency at cutoff, and $λc$ the excitation wavelength at cutoff. This equation permits one to put . The amplitudes of the field intensities are chosen such that $Q$ equals Planck's constant $h$⁠. Applying the mass-energy relation, the term $hμc = hc/λc = m0c2$ may be interpreted as the “potential” (= rest) energy of the quantum $hμ$ and $m0$ as the rest mass. The momentum $p$ associated with $hμ$ is found to be $p = h/λg$⁠, $λg$ being the wavelength along the guide. When radiant energy passes from one guide into another one of different cross-sectional structure via a tapered piece of wave guide, a partial conversion of “potential” to “kinetic” energy, and vice versa, occurs along the tapered section. In the case of TEM waves, which, however, can only be supported in a guide of infinite transverse dimensions, the group velocity becomes $c$ and the rest mass zero.