# What would be the associated wavelength of the particle if its velocity is zero?

What will be the wavelength of a particle whose velocity is zero?

According to de Broglie's hypothesis, then the wavelength would become infinite as the momentum is zero. But, I think for a stand still particle, its particle nature should be more dominant, as at that moment it is highly localized.

## 1 Answer

To the contrary, the slower the particle moves, the more its wavelike properties show up. Compare e.g. electron in an atom, where its energy is at its lowest, with an electron flying out of a CRT. In the former case we need quantum mechanics to describe its motion (it's where QM originates), while in the latter case classical mechanics is sufficient.

So the wavelength becoming infinite for a resting electron is a completely consistent result. And it's also consistent with Heisenberg's uncertainty principle: momentum is exactly defined while position is completely undefined.

• Yeah! In microscopic system this seems to make quite sense. But I'm confused for macroscopic systems. If a ball is at rest, it's position and momentum both seems to be well defined, which I can't corelate with QM. Please explain where I'm going wrong – user137271 Nov 29 '16 at 16:40
• @user137271 Indeed, they seem to be well defined. But they are only well defined on the human scale. If you actually measure its position and momentum and multiply the measurement errors (uncertainties), you'll get much larger product than $\sim\hbar$ (many orders of magnitude larger!). Thus, it still doesn't contradict the uncertainty principle. – Ruslan Nov 29 '16 at 16:46
• But shouldn't the "wavy" nature of the ball be detectable by the human eye becoz, from the formula we can see that the wavelength is tending towards infinity as velocity is approaching zero?? – user137271 Nov 29 '16 at 16:53
• @user137271 to say that velocity of classical object approaches zero, you must measure the velocity very precisely. But even then, you don't take into account that different parts of the object move with different velocities due to thermal motion, being hit by gas molecules, interacting with light etc. – Ruslan Nov 29 '16 at 18:50