In particle-particle scattering, one is said to be a target and the other is moving at some velocity. But since we know with small uncertainty the position of the target particle, is the momentum and hence speed uncertainty huge? Can it be measured at one instant to be zero and another instant to approach the speed of light? I'm still not well versed in quantum mechanics, but I'm just curious.


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Even if the momentum uncertainty is large, the "speed uncertainty" can be small if the particle has a large mass. In practice, a static potential in quantum mechanics usually represents an interaction between a light "probe" particle and a heavy "target" particle. In classical physics, you would expect the probe particle not to impart much momentum to the target if the target's mass is much larger than the probe's mass.

In quantum mechanics, we can consider the target to be approximately stationary so long as its position uncertainty is smaller than the wavelength associated than the incoming probe particle. So, $\Delta x_{\rm target} \ll \frac{\hbar}{p_{\rm probe}}$. Then from the uncertainty principle, the target has a momentum uncertainty of around $\Delta p_{\rm target} \geq \frac{\hbar}{2 \Delta x_{\rm target}} \gg p_{\rm probe}$.

If the target particle has mass $M$ and the probe particle has mass $m$, then using $p=mv$, this translates into \begin{equation} \Delta v_{\rm target} \gg \frac{m}{M} v_{\rm probe} \end{equation} Since $m/M$ is a tiny number by assumption, this bound is actually very weak. $\Delta v_{\rm target}$ can even satisfy this bound without being larger than $v_{\rm probe}$. So the "velocity uncertainty" $\Delta v_{\rm target}$ need not be very large at all.

Now, one should also point out that the notion of a particle's "velocity" in quantum mechanics is a bit tricky, since a particle does not have a well defined trajectory. But, loosely speaking, we can think of the velocity as telling us how far we would expect the wave function to spread in a unit time interval, assuming the particle was well localized initially. The fact that the velocity of the target can be very small, consistent with the uncertainty principle, is telling us that we can ignore the spread of the target's wavefunction to a good approximation, for at least some interval of time which is large on the scale at which the probe's wave function evolves. Exactly how good this approximation is will depend on the details of $m/M$, but in many situations (for example, electron crystallography), it is a perfectly fine approximation to treat the target as a static potential.


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