For which of the following particles will it be most difficult to experimentally verify the de-Broglie relationship?

The de-Broglie relationship states that the wavelength $$\lambda$$ associated with a moving particle is given by $$\lambda = \frac{h}{p}$$, where $$h$$ is Planck's constant and $$p$$ is the momentum of the particle. The momentum $$p$$ is equal to $$m \times v$$, where $$m$$ is the mass and $$v$$ is the velocity. Therefore, $$\lambda = \frac{h}{m v}$$.

To experimentally verify the de-Broglie relationship, we need to observe wave-like behavior such as diffraction or interference. This requires the de-Broglie wavelength to be comparable to the size of the slits or obstacles in the experiment, typically on the order of atomic dimensions (about $$10^{-10}$$ m). If the wavelength is too small, the wave effects become negligible and difficult to detect.

The wavelength $$\lambda$$ is inversely proportional to the mass $$m$$ for a given velocity $$v$$. Therefore, heavier particles will have shorter wavelengths, making it harder to observe their wave nature.

Now, comparing the masses of the given particles:

• An electron has a mass of approximately $$9.1 \times 10^{-31}$$ kg.
• A proton has a mass of approximately $$1.67 \times 10^{-27}$$ kg, which is about 1836 times the mass of an electron.
• An $$\alpha$$-particle (helium nucleus) has a mass of approximately $$6.64 \times 10^{-27}$$ kg, which is about 4 times the mass of a proton.
• A dust particle is macroscopic. Even a very small dust particle, say with a diameter of 1 micrometer and density of $$2000$$ kg/m³, has a mass of about $$10^{-15}$$ kg. This is many orders of magnitude larger than subatomic particles.

Since the dust particle has the largest mass, its de-Broglie wavelength will be the smallest for the same velocity. For example, assuming the same velocity $$v$$:

$$\lambda_{\text{electron}} = \frac{h}{m_{\text{e}} v}$$

$$\lambda_{\text{dust}} = \frac{h}{m_{\text{dust}} v}$$

Given $$m_{\text{dust}} \approx 10^{-15}$$ kg and $$m_{\text{e}} \approx 10^{-30}$$ kg (approximately), the ratio is:

$$\frac{\lambda_{\text{dust}}}{\lambda_{\text{electron}}} = \frac{m_{\text{e}}}{m_{\text{dust}}} \approx \frac{10^{-30}}{10^{-15}} = 10^{-15}$$

Thus, $$\lambda_{\text{dust}}$$ is about $$10^{15}$$ times smaller than $$\lambda_{\text{electron}}$$. Even if the dust particle is moving very fast, its wavelength remains extremely small. For instance, with typical velocities, $$\lambda_{\text{dust}}$$ might be around $$10^{-25}$$ m or smaller, far below atomic scales ($$10^{-10}$$ m), making it impossible to detect with current technology.

In contrast, electrons, protons, and $$\alpha$$-particles have been used in experiments like electron diffraction or Rutherford scattering to confirm wave-particle duality because their wavelengths are measurable.

Hence, the dust particle poses the greatest experimental challenge for verifying the de-Broglie relationship.

So, the answer is Option C.