Does velocity make sense if it is small relative to the de Broglie wavelength?

Physics Asked by Allure on September 21, 2020

What does it mean if a particle has, say, de Broglie wavelength of $100m$ and a velocity of $1 m/s$? Is it even possible to have such a setup? I don’t see why not, since we can always slow the particle down. But if it is possible, what can we say about the particle’s position after $1s$?

Does it actually make sense to say the particle had a velocity of $1 m/s$? After all, by quantum uncertainty, we don’t know where the particle is within that $100m$, so any statement of its "displacement" would have error bars of $pm 50m$, making the velocity meaningless. But if that’s the case, then for some "minimum mass", the de Broglie wavelength would be of order light years, and then speed would cease to be meaningful since it would have error bars $pm c$.

I’m hoping to get some intuition for this scenario where the de Broglie wavelength is large relative to the velocity (knowing full well they aren’t directly comparable since they have different units).

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