Understanding the De Broglie Wavelength of Particles with the Same Kinetic Energy

In this article, we will explore the fascinating relationship between the de Broglie wavelength and the kinetic energy of particles. Specifically, we will discuss how a proton, an alpha particle, and a beta particle, all having the same kinetic energy, compare in terms of their de Broglie wavelengths.

The De Broglie Wavelength Formula

The de Broglie wavelength is a fundamental concept in Quantum Mechanics. It relates the wavelength of a particle to its momentum. The formula for the de Broglie wavelength is given by:

λ h / p

where:

λ is the de Broglie wavelength h is Planck's constant, approximately 6.626 × 10-34 Js p is the momentum of the particle.

Momentum and Kinetic Energy

Momentum, p, can be expressed in terms of kinetic energy, K, through the following equation:

K p2 / (2m)

By rearranging this equation, we can express momentum as:

p √(2mK)

Substituting this expression for momentum into the de Broglie wavelength formula, we get:

λ h / √(2mK)

From this equation, it is clear that the de Broglie wavelength (λ) is inversely proportional to the square root of the mass (m) of the particle, provided the kinetic energy (K) is constant.

Comparing Particles with the Same Kinetic Energy

Let's now compare a proton, an alpha particle, and a beta particle, all having the same kinetic energy, in terms of their de Broglie wavelengths.

Proton

Mass of the proton, mp ≈ 1.67 × 10-27 kg

Alpha Particle

An alpha particle is composed of 2 protons and 2 neutrons. The mass of the alpha particle, mα, is approximately 4 times the mass of a proton:

mα ≈ 4 × 1.67 × 10-27 kg ≈ 6.68 × 10-27 kg

Beta Particle (Electron)

The mass of an electron, mβ, is much smaller:

mβ ≈ 9.11 × 10-31 kg

Conclusion

Given that the de Broglie wavelength is inversely proportional to the square root of the mass, and the mass of the beta particle (electron) is the smallest among the three particles, it follows that the beta particle will have the largest de Broglie wavelength. Therefore, the order of wavelengths from largest to smallest is:

λβ (beta particle) λp (proton) λα (alpha particle)

Conclusion: The beta particle (electron) has the largest de Broglie wavelength among the three particles when they all have the same kinetic energy.

Further Considerations

It's important to note that achieving the same kinetic energy as a proton requires the electron to travel at relativistic speeds, which would introduce additional complexities. However, the fundamental principle remains that the smaller the mass, the larger the de Broglie wavelength for a given kinetic energy.