It is well known that for a free particle, described by $H=\hat{p}^2/2m$,
$\hat{p}_{x}(t)= constant$ (similarly for other components of momentum). Meanwhile, $\hat{x}(t)$ is not a constant, being proportional to $\hat{p}_{x}(0)t/m$ (and similarly for other components) .
An operational interpretation of this result could be this: We prepare an ensemble of setups, in each of which we put a set of position detectors along a direction which we call the $x$-axis, at a given $t>0$, and send in a free particle of mass $m$ from a source at origin. We observe a frequency distribution in the measured position observable, consistent with the expectation value of position that we calculate. Meanwhile, we could infer the momentum that the particle has in each position detection event from the relative position of the position detector from the source at origin and the time taken to arrive. Again, we find a result consistent with our calculation of the expectation value for momentum. I.e, we see a constant value for the $x$-component of the momentum, from each $x$-component position detection events.
Now, I was working on a problem-(The problem I considered in it's 1-d version is H=xp+V(x). The first term is the Von Neumann position measurement interaction term, which he mentions in the final chapter of his well known quantum book. The x is the position observable corresponding to the system and the p is the momentum observable corresponding to the probe. V(x) is some potential that I considered to affect only the system. I assumed x and p to commute as they correspond to different particles. Solving the Heisenberg equation for this setup gives the desired result. I neglected the kinetic energy terms.) -that had a Hamiltonian which results in an equations of motion that was roughly the reverse of a free particle in the sense that $\hat{x}(t)=constant$(and has a small variance) and $\hat{p}_{x}$ is time dependent. This equations of motion seem to mean that the particle has a sharply localized wave packet in position space for all time, but the momentum shows a large (I am being rough here) uncertainty/distribution in its measured values. I was looking for a simple experiment that could instantiate this result, but could not find a satisfactory one. So any help is well appreciated.
I initially did think of the single-slit experiment, where we have have a narrowing of the position space wave packet as a result of confining the particle to the single narrow slit. We know that the state of the particle consequently acquires a large uncertainty in momentum that could be verified by putting a screen in front of the single slit. This supposed operational example for the equations of motion that I got does not appear satisfactory because, it appears that one cannot claim that the uncertainty in position for the wave-function just before the detection in the screen, was the same as the uncertainty in position (which was very small in magnitude) while the particle was confined to the narrow slit. i.e $\hat{x}(t)=constant$ is not strictly satisfied.
Another operational method I thought of is to do a continuous position measurement that indeed keeps $\hat{x}(t)=constant$. But then, how can we measure the momentum of a particle that is for all time at a given spot (and importantly, find a large momentum uncertainty)?
Q&A (4871)
