Question

All other things being equal, which would be more likely to exhibit its wave nature—a proton or an electron—and why? By making something unequal, how could you “compensate,” so as to make one as wavelike as the other?

Short answer

The electron is more wavelike than the proton, in accordance with the de Broglie relation, although they both display the same wave properties if the electron moves around 2000 times more slowly than the proton.

Step-by-step solution

The wavelength of a particle

The wavelength of a particle is given by$\mathbf{}\mathit{\lambda }\mathbf{=}\frac{\mathbf{h}}{\mathbf{p}}\mathbf{=}\frac{\mathbf{h}}{\mathbf{m}\mathbf{v}}$ .

Explanation

Because of the inverse connection between wavelength and momentum provided by the de-Broglie equation, tiny objects are often more likely to display wave-nature. The concept is that in order to notice the wave characteristics in measurements, the objects must have more strong wave characteristics, which is the case for tiny particles like the electron. However, the wavelength is so narrow and difficult to notice for large objects.

According to their mass ratios, the proton will have a momentum 2000 times greater than the electron if we assume that both the electron and the proton are travelling at the same speed. However, we may deduce that the wavelength of the proton will be 2000 smaller than the wavelength of the electron.

We return to the de-Broglie equation to find that the wavelength decreases with increasing velocity in order to account for the change in wave nature. As a result, both particles will exhibit the same wave characteristic whether the proton can move 2000 times slower than the electron or, conversely, if the electron can move 2000 times faster than the proton.