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 

\(K = \frac{1}{2}mv^2\)

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

\(\lambda =\frac{h}{p}\)

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

\(\Rightarrow \lambda =\frac{h}{p}\)

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.