Question

What is the ratio of the de Broglie wavelengths for an electron and a proton if both have equal kinetic energy?

Hint:

For two particles with the same kinetic energy, the ratio of their de Broglie wavelengths is inversely proportional to the square root of their masses.

Answer 1

At the same kinetic energy, the lighter particle always ends up with the longer de Broglie wavelength, because wavelength shrinks as momentum grows and a lighter mass needs less momentum to carry the same energy. Since the electron is far lighter than the proton, its wavelength works out to \( \sqrt{m_p/m_e} \) times longer than the proton's wavelength, i.e. \( \lambda_p/\lambda_e = \sqrt{m_e/m_p} \).