The de Broglie wavelength of a particle is given by the equation:
λ = h / p,
where λ is the de Broglie wavelength, h is Planck's constant (approximately 6.626 x 10^(-34) J·s), and p is the momentum of the particle.
The momentum of a particle can be calculated using the equation:
p = √(2mE),
where m is the mass of the particle and E is the kinetic energy gained by the particle.
In this case, we have a proton and an alpha particle accelerated through the same potential difference. Since the potential difference is the same, the kinetic energies gained by the proton and the alpha particle will be equal.
Let's denote the mass of the proton as m_p and the mass of the alpha particle as m_alpha.
The ratio of their de Broglie wavelengths can be calculated as follows:
λ_proton / λ_alpha = (h / p_proton) / (h / p_alpha) = p_alpha / p_proton.
Using the equation for momentum, we can write:
λ_proton / λ_alpha = (√(2m_alphaE) / √(2m_pE)) = √(m_alpha / m_p).
Therefore, the ratio of the de Broglie wavelengths of a proton and an alpha particle accelerated through the same potential difference is equal to the square root of the ratio of their masses:
λ_proton / λ_alpha = √(m_alpha / m_p).