Two-Slit Experiment And Varying Electron Momentum

[NaiveTay]

There seems to be two divided approaches in how the uncertainty principle is explained, but they seem to be explaining two different things.

The first, more intuitive explanation of the limits imposed by quantum mechanics goes something like: in order for a measurement to be made, we have to affect the thing we wish to measure, otherwise we clearly can't retrieve any information from that so called thing. As a result, the mere act of measuring disturbs the information we were trying to observe.
Because of the inherent wave-like properties of all forms of energy, and the combined fact that increasing momentum decreases wavelength, we're left with an inescapable trade off. We can either keep the momentum of our probe (the photon, electron, etc. we're using to make the observations) low, so as to retain more of the original information we're trying to measure, but as a result lose the ability to measure accurately with our now spatially expanded probe; or we can do the reverse and have a probe with a very high degree of accuracy but because of the increased momentum we inevitably change the desired information much more severely.

That's all well and good, but it appears to be missing something. This brings us to the other explanation of quantum theory, the two-slit experiment.
Sending an electron (or any other particle) at a barrier with two slits closely together, we end up with an interference pattern on the sensor we place behind this barrier. Of course, the best part is the result that sending the electrons through one at a time, the interference pattern remains. The particle interferes with itself, it "goes" through both slits.

These, to me, seem to be two related, but different things.

The first just explains the interaction of two particles. The second, however, seems to only be the results of how a particle moves through space.
The first is just the fact that we are made of stuff, and we're trying to observe different stuff, so we have to use even more stuff to interact with what we're trying to observe in order to bring that information back to our stuff-built sensors. But that, in and of itself, doesn't lead to the conclusions of quantum tunneling and the observed interference pattern of singly shot electrons. It just says that when things interact they affect each other, i.e., the act of measuring changes the system we are measuring.Now the question I have that I believe will shed some light on this apparent divide, is what happens to the interference pattern if we increase the velocity of the electrons we shoot at the two slits? Does this decrease the area of the interference pattern like the uncertainty principle would predict?
In other words, does the law for how particles interact also apply to a particle which is traveling through empty space (the area of space of the two slits) and manifest according interference due to this law before it hits the sensor? Or does a particle interfere with itself independent of the uncertainty principle?

Your insights will be very much appreciated.

 

[vanhees71]

I cannot repeat it often enough. The usual Heisenberg-Robertson uncertainty relation is about the impossibility to PREPARE a particle in a state that represent the particle to have a sharp position in one direction and a sharp momentum in the same direction, $\Delta x_j \Delta p_k \geq \hbar \delta_{jk}/2$. It does not prevent you from measuring position or momentum as accurately as you like.

In fact, to check the uncertainty relation you need an ensemble of equally prepared particles and do a position measurement with a much larger accuracy than $\Delta x$ and an ensemble with the same properties to make a momentum measurement with a much larger accuracy than $\Delta p$.

 

[NaiveTay]

I cannot repeat it often enough. The usual Heisenberg-Robertson uncertainty relation is about the impossibility to PREPARE a particle in a state that represent the particle to have a sharp position in one direction and a sharp momentum in the same direction, $\Delta x_j \Delta p_k \geq \hbar \delta_{jk}/2$. It does not prevent you from measuring position or momentum as accurately as you like.

Yes, that makes sense. We never know the initial state of a particle or of our apparatus, so we can't ever find the exact value for the final state because we need to compare initial to final in order to find the change.
But that's not my question. The world could be completely classical and those limits would still be imposed.
The two-slit experiment shows more than that. A particle's position and momentum aren't just unknowable, even more, those two properties aren't independent of each other in any respect.

If the problem were just our ignorance of the situation, a particle would only go through one slit and it wouldn't interfere with itself, though where it would hit would still be indeterminable. But a single particle is affected by having both slits open, so that tells us something of the intrinsic properties beyond just the fact that we can't know where they exist and how they are moving.
So that brings me back to my question. If an entity, which we call a particle/wave, doesn't have exclusively existing properties we call position and momentum, then the way a particle interferes with itself should change if we vary one of these properties. But I haven't seen the experimental evidence of this so I don't really know. I'm askin about modifications of the two-slit experiment and how they affect the results... The equations more or less make sense to me.

 

[vanhees71]

Well, also in classical physics position and momentum are not independent from each other, because by definition in theoretical physics the momentum of a system is the operator that generates spatial translations. In QT from this the commutator relations for the corresponding operators follow, and from the commutator relation the Heisenberg uncertainty relation. Nevertheless in classical mechanics the particle always has well defined position and momentum. Even more, in classical mechanics all observables always take definite values, no matter whether you know them or not. That's the most unfamiliar property of quantum physics compared to classical theory. Quantum theory is probabilistic, and which observables take determined values is dependent on the state the system is prepared in, while classical physics is deterministic and all observables always take determined values.