# Does momentum space have a speed limit?

In ordinary $xyz$ space, the maximum velocity of propagation for mass-energy and/or information is $c$.

So, my question: Is there also a maximum velocity of propagation in momentum ${p_x}{p_y}{p_z}$ space, one that would for example place a maximum limit (not necessarily approached) on how quickly an electron can fall from the top of a conduction band into a newly opened hole at the bottom of the band?

If not... well, why not? It's easy to say "no," but I'm not aware of any actual study of such a question. Given some of the remarkable symmetries and mathematical links between the $xyz$ and ${p_x}{p_y}{p_z}$ spaces, I suspect that some sort of specific theoretical answer should be possible.

• the velocity of propagation in momentum space is actually the same as force: $\dot p=f$, by definition. In principle, the force can be arbitrarily high (e.g., $f\propto 1/r^2$ can take any value as $r\to 0$). – AccidentalFourierTransform Jan 18 '16 at 22:24
• Nice point, one that makes me think "I should have thought of that!" (Cool name BTW, apt for this question.) So my question is equivalent to "Is there a maximum acceleration regardless of the magnitude of the force applied?"... which sounds like the situation when approaching $c$... hmm! – Terry Bollinger Jan 18 '16 at 22:37
• Electrons "falling" from one band to another are a quantum mechanical phenomenon that is subject to a time-energy "uncertainty relation". It doesn't really make sense to look at them with the tools of kinematics. – CuriousOne Jan 19 '16 at 4:24
• Good point for consideration, thanks! I gave that one mostly because motion is such a tricky issue in momentum space, and that is one that has meaning. I am... genuinely unsure?... whether such a band drop, which is a quite high energy event (X-ray photon emission for silver) qualifies as a purely quantum event in momentum space, or instead has a kinematic component. I suspect the latter because those high energies which would not be typical of something falling under the auspices of time-energy uncertainty. But in any case, my intent was to give that only as an example, not the defining case. – Terry Bollinger Jan 19 '16 at 6:06

For a transition like electron-hole recombination, its probability is linear with time with some characteristic time scale that depends on the system: the probability of having a transition between $0$ and $dt$ is $\frac{dt}{\tau}$. So there is a non-vanishing transition probability at arbitrarily small times.
In momentum space, the evolution of a wavepacket looks something like this: it is nonzero near a point $\vec p$ for a while, until it scatters and jumps to some other $\vec p'$. I don't see how $c$ comes into this.