Toward a Complete Wave-Particle Duality: Do Matter Waves Have Inertia?

A quaternionic mass defining longitudinal mass () and transverse mass () of a particle is introduced. The longitudinal and transverse masses represent the mass of the particle and matter wave accompanying it, respectively. A massive particle is found to have a bigger longitudinal mass than transverse mass. The particle nature due to the transverse mass of a particle is described by a scalar wave satisfying the Klein-Gordon equation. The transverse mass vector is found to be along the direction of the particle’s velocity (). A particle of an equal longitudinal and transverse mass travels at speed of light in vacuum. The transmission of matter wave energy is found to be opposite to the particle velocity. The momentum of the particle () and its matter wave () are equal. The direction of the transverse mass is found to be along the direction of the particle’s velocity.