Question

If a proton and an electron have the same speed, which has the longer de Broglie wavelength? Explain

Short answer

Electron has longer de Broglie wavelength

Step-by-step solution

Expression for de Broglie wavelength

The de Broglie wavelength is given by

$\mathit{\lambda }\mathbf{=}\frac{\mathbf{h}}{\sqrt{\mathbf{2}\mathbf{m}\mathbf{K}}}$

Where, m is the mass of the particle and K is the kinetic energy of the particle.

Identification of longer de Broglie wavelength

From the above formula is it clear that the de Broglie wavelength is a function of mass. Replace Kinetic energy by

$K=\frac{1}{2}m{v}^2$

We get

$\lambda =\frac{h}{\sqrt{m^2{v}^2}}$

Since the velocity is the same for both the particles, we get the wavelength is higher for the lighter particle which here is the electron

Hence Wavelength of the electron is longer.