De Broglie Hypothesis

To determine the relationship between the wavelengths associated with an electron, a proton, and a deuteron all moving at the same speed, we utilize the de Broglie wavelength formula:

$ $

where is the wavelength, is Planck's constant, is the mass of the particle, and is the velocity. Given that the speed is the same for all three particles, the wavelengths are inversely proportional to their masses:

1.

2.

3.

Where and are the masses of the electron, proton, and deuteron respectively. Given:

- The mass of an electron is the smallest.

- The mass of a proton is larger than the mass of an electron.

- The mass of a deuteron (approximately twice the mass of a proton) is the largest.

Therefore, since the masses follow , the wavelengths satisfy . Hence, the correct relation between the wavelengths is: