Yes, a fast-moving massive particle, such as an electron or a proton, can display wave-like properties. This phenomenon is known as matter wave or de Broglie wave, named after the French physicist Louis de Broglie who proposed the concept.
According to de Broglie's hypothesis, particles with mass, momentum, and energy also exhibit wave-like characteristics. The wavelength associated with a moving particle, often referred to as its de Broglie wavelength (λ), is inversely proportional to its momentum (p):
λ = h / p
where λ is the wavelength, h is the Planck constant (a fundamental constant in quantum mechanics), and p is the momentum of the particle.
The de Broglie wavelength implies that even macroscopic particles, like electrons or protons, can exhibit wave-like behavior under certain conditions. However, the wave nature of such massive particles is typically only noticeable on very small scales, such as in experiments involving atoms or subatomic particles.
The mathematical formalism that describes matter waves is called wave mechanics or quantum mechanics. It provides a framework to describe the behavior of particles, including their wave-like properties. In quantum mechanics, particles are described by wave functions, which are mathematical functions that encode the probabilities of different outcomes when measuring the properties of the particle.
The wave-particle duality, demonstrated by the wave-like behavior of particles, is a fundamental concept in quantum mechanics and has been extensively verified through various experiments and observations.