If alpha, beta and gamma rays carry same momentum, which has the longest wavelength?

CONCEPT:

De Broglie wavelength: It is a wavelength expressed in all the objects in quantum mechanics which determines the probability density of finding the object at a given point of the configuration space.

The de Broglie wavelength of a particle is inversely proportional to its momentum.

The kinetic energy of the particle is 

Where K = kinetic energy, m = mass of particle, v = velocity of particle.

The momentum of the particle is 

P = m v 

Where p = momentum of the particle,

The de Broglie wavelength is given by

Where λ = de Broglie wavelength, p = momentum of the particle.

EXPLANATION:

Alpha rays: Alpha ray, also called alpha particles alpha ray, contains two protons and two neutrons bound together into a particle just like a helium-4 nucleus.

Beta rays: Beta rays, also called beta particles or beta radiation, maybe a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus during the process of beta decay.

Gamma rays: A gamma-ray, or gamma radiation, is penetrating electromagnetic radiation arising from the radioactive decay of atomic nuclei.

The de Broglie wavelength is given by

The wavelength of this wave is determined by the momentum of the particle.

If p is the momentum of the particle, the wavelength of the wave associated with it is λ = h/P

Where h is Planck's constant,

Since, it is given that, alpha, beta, and gamma rays carry the same momentum, so they will have the same wavelength.