Particle in a potential channel (waveguide-optical analogy)

The motion of a particle in a potential channel (two-dimensional well without a potential gradient along the third coordinate) is characterized by a discrete set of allowed modes or travelling Ψ waves. A paraxial potential channel can transmit the initial distribution of the Ψ function to channel cross sections some distance from the entrance. The dispersion equation for a channel having transverse dimensions of the order of the Compton wavelength is the same as the free motion equation, the eigenvalue of the lower channel mode being identified with the rest energy of the particle.