Do the same experiments yield different results due to the principle of uncertainty?

Question

When thinking about small particles and their uncertainity, I've allways rather seen them being all over the place rather than randomly changing location. I would think that, in the same time, you'd always have same result, have you tried to do something to them (eg. shoot another particle at them).

However, in certain situations, like the evaporation of a black hole, the particle just either is here or it is not.

This makes me wonder: if you try something with uncertain particles, does every experiment end up differently because, event though all conditions are the same? (The last requirement seems to be unfeasible so consider it just a thought experiment).

Similarly, if there was a separated Big bang would the other universe look different than ours because particles ware randomized differently?

_{Out of the question, I must say that I find it very confusing for particles to really have random locations. Wasn't there ever a suspicion that their location (as well as the probability of it) has a rule we can find out?}

Answer 1

Strictly speaking, to see the uncertainty principle at work, you need to do two or more experiments, or at least "one experiment" that makes at least two "measurements". Here, by "measurement" I mean the gleaning of a result modelled by the application of an observable to a quantum state. The uncertainty principle talks about bounds on the uncertainties of two measurements.

The random variation you see in repetitions of the same, one-observable experiment is simply the forcing of a quantum system's state into one, seemingly random, eigenstate of the observable that models the measurement. It's a random projection of the state onto the basis of eigenvectors of that observable. What actually happens and how it happens, i.e. whether it really is a "collapse" of a real state or the updating of a distribution analogous to the classical notion of switching to a different conditional probability distribution when one has new information on a system, is still an open question and is called the quantum measurement problem

The uncertainty principle works like this: you have two, noncommuting observables, say position and momentum. You make the first, position measurement. It's result is random, and it forces the quantum state of the observed particle into an eigenstate of the position observable with an exact position value. Because the eigenstates of the momentum observable are not the same as those of the position's (indeed two operators, or matices, commute if and only if they have the same eigenvectors), the second measurement forces the systems state into a second, different state and the projection is random, with a probability distribution given by the square of the magnitude of the position eigenstate's components with respect to the momentum co-ordinates. If the two observables commute, then the eigenvectors are the same and the second measurement will be exact and wholly set by the first.

You allude to the idea of hidden variables when you talk about an experiment's yielding the same result if "everything" were exactly the same. That is, you seem to be saying that there are hidden variables we don't know about. There is a version of quantum mechanics that takes just this approach, namely Bohmian mechanics but it is nonlocal, in the sense that it implies causal relationships between two events that are spacelike in separation, i.e. the causal relationship would imply faster than light signalling. Bohmian mechanics is therefore the most difficult interpretation to reconcile with relativity, and is thus not favoured by mainstream physics.

Answer 2

When thinking about small particles and their uncertainty, I've always rather seen them being all over the place rather than randomly changing location. I would think that, in the same time, you'd always have same result, have you tried to do something to them (eg. shoot another particle at them).

You have to define "particle". When going to dimensions below a nanometer the term "particle" is not the classical particle as a billiard ball. Depending on the experiment/observation it will give an answer "particle" or "wave". This does not mean that it is spread out all over the place. It only means that using quantum mechanical solutions for the problems under consideration we do not get trajectories or orbits, but just probabilities of how probable it is to find the electron ( as an example) at ( x,y,z,t). This means that to get any reliable value we have to accumulate statistics.

The best way is to contemplate the two slit experiment with single electrons. In this experiment you throw an electron against two narrow slits with small separation .

The single dots in the upper picture are single electrons which have crossed the two slits and then interacted with the screen as a dot, a given ( x,y,z). After statistical of many throws, an accumulation gives the probability distribution ( which is the square of the quantum mechanical solution for the problem, the state function), and we see the wave nature appearing. Also an uncertainty of position on average ( which would allow us to know the momentum with great accuracy)

This makes me wonder: if you try something with uncertain particles, does every experiment end up differently because, event though all conditions are the same? (The last requirement seems to be unfeasible so consider it just a thought experiment).

As you see in the experiment above, yes, each single outcome has an uncertainty. It is the accumulated distribution that gives the numbers to compare with predictions.

As for the Big Bang, our observations lead us to the observable universe and we have a model for it. In the accepted theoretical model there is only one BB.

Out of the question, I must say that I find it very confusing for particles to really have random locations. Wasn't there ever a suspicion that their location (as well as the probability of it) has a rule we can find out?

Hidden variable theories have problems with the Lorenz invariance which is something that all of our experimental observations say holds without exception.

Answer 3

I think perhaps you are confusing uncertainty with distribution. The principal of uncertainty states that if you attempt to measure position, your knowledge of the momentum decreases (and vice versa). This is due to the fact that our current theory of the relationship of position and momentum (and consequently time and energy) is by Fourier Transform. Fourier transforms are only exact when the function which you are transforming is continuous, and thus when dealing with Fourier analogs, there will always be some error inherent to the calculations. This manifests itself in the uncertainty principle when applied to, for example, position and momentum. That doesn't mean that each individual particle doesn't have a distinct position and momentum at any given instant (and in fact, a relatively recent publication explained here may imply that they do). Thus, although there is variation in our measurements, a properly delineated experiment won't have much in the way of variation in outcome due to uncertainty.

Part of the difficulty when dealing with measurements involving quantum interactions though, is that when we measure we measure a single instance out of an ensemble of particles. This leads to a probability distribution, since we can't repeat the exact same circumstances (yet) on a particular basis. What we do find is that when we measure for any particular thing, we build the same probability distribution. The probability distributions are distinctly different than uncertainty in that the particles measured in each experiment are different particles, and the best we can do is create a statistical likelihood for certain outcomes given known initial conditions.