The de Broglie wavelength of a particle is given by the formula λ = h / p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle.
The momentum of a particle is related to its kinetic energy (KE) through the equation p = √(2mKE), where m is the mass of the particle.
Since the kinetic energy is the same for the electron, the alpha particle, and the proton, we can compare their de Broglie wavelengths based on their masses alone.
The electron has the smallest mass among the three particles, followed by the proton, and the alpha particle has the largest mass. As the mass increases, the de Broglie wavelength decreases.
Therefore, the electron would have the shortest de Broglie wavelength among the electron, alpha particle, and proton when they all have the same kinetic energy.