Is the Uncertainty Principle Statistical? And if so

[nonequilibrium]

Hello,

I'm (only) receiving an introduction to Quantum Physics atm, and today our professor argued that the uncertainty principle doesn't have a definite form: the right hand side of the inequality is dependent on the confidence measure/certainty you choose, in other words 95%, 99% certainty, ... So is this true? This would make the general statement $$\Delta x \Delta p = O(h)$$.

The weird thing: if it is statistical, it means that it can happen that your measurement was more exact than any specific certainty level that you could expect beforehand. I do understand that you can't do it repeatedly, just like the arrow of time is statistical which doesn't mean we can make all the gas in a room collect in a corner any time we want (without expenditure of work...), but the thing is: I remember having read in more than one book that quantum mechanics (or at least certain intepretations of it) take the UP to mean that x and p are actually not defined to an arbitrary certainty level at any given time. But the statistical version of the UP would imply that x and p are always defined to an arbitrary precision, because there always exists a certain chance the uncertainty is smaller than any specific right hand side (chosen beforehand) of the UP.

 

[DrChinese]

I remember having read in more than one book that quantum mechanics (or at least certain intepretations of it) take the UP to mean that x and p are actually not defined to an arbitrary certainty level at any given time.

The experimental evidence seems to support this quite strongly, although as you say it is somewhat dependent on the interpretation. See EPR (1935) and Bell/Aspect for more information; the short story is that entangled particle pairs (usually called Alice and Bob) respect the HUP (UP) even when spacelike separated. You might have otherwise expected to be able to "beat the HUP" by measuring x on one and px on the other. But that doesn't work!

 

[diazona]

The uncertainty principle is a statistical equation, but not in the way your professor is saying. Here's what it really means: for any particular quantum state, if you multiply the standard deviation of the probability distribution function (PDF) of position by the standard deviation of the PDF for momentum, you'll get a result that is at least $\hbar/2$.

In reality, we can't measure a quantum state (or the associated probability distribution function for an operator) directly. What we can do instead is perform a large number of measurements on systems which start out in the same state, and the distribution of the results we get will approximate the PDF. So in practice, you could start with a large number of quantum systems in the same state, and measure the position of half of them and measure the momentum of the other half. Then compute the standard deviation of the positions you measured and the standard deviation of the momenta you measured, and multiply them together. The result should be at least $\hbar/2$.

Of course, since measurement is a random process, it's possible that your particular set of results might give you a lower value for the product, and perhaps that's what your professor is referring to. But that would be a consequence of only having made a finite number of measurements. It's just like if you roll a die a finite number of times, it's possible to get a 6 every time.

Bottom line: the uncertainty principle is really a statement about the theory, and in that sense it is perfectly definite.

 

[nonequilibrium]

Thank you both for posting.

The UP often gets argued as being a physical consequence, graphically: for knowing its momentum, you have to let some time pass, making its position at that moment less well-defined due to movement, etc. And I also remember being argued that the UP shows the particle-wave-duality as one factor is often associated with a wave and the other with a particle, implying that they can't we both 100% at the same time (although I must say both x and p seem to be particle-like...) But diazone describes an experiment where you take half the group for x-measurements and the other half for p-measurement. So is the afore-mentioned description of the UP -where loss of information of one variable is the result of trying to measure the other- simply deadwrong? (because that's also how we saw it in class: he looked at a particle that went through a narrow slit (giving a small uncertainty in y-position) after which (due to wave-like behavior) diffraction occurred making its momentum more undetermined in the y-direction.

On the other hand, the way diazona describes that measuring process does show that the UP in itself is independent of the intepretation of the being-100%-defined of x and p at the same time. This is good, as it resolves my original question. Now the only problem lies in the paragraph above.

 

[diazona]

The UP often gets argued as being a physical consequence, graphically: for knowing its momentum, you have to let some time pass, making its position at that moment less well-defined due to movement, etc.

That view I would say is wrong. It's not actually true that you have to let some time pass to determine momentum - at least, not in theory. At any single instant of time, any physical system is in some state. Classically, the state of a system is defined by the momentum and position of each particle; or in quantum mechanics, the state is something more fundamental from which position and momentum (or rather, their PDFs, expectation values, uncertainties, etc.) can be calculated. But either way, the state of the system at a particular instant gives you all the information you can know about both position and momentum, and there's no need to let any time elapse.

And I also remember being argued that the UP shows the particle-wave-duality as one factor is often associated with a wave and the other with a particle, implying that they can't we both 100% at the same time

I would say that's just a misappropriated analogy: wrong at worst and useless at best.

So is the afore-mentioned description of the UP -where loss of information of one variable is the result of trying to measure the other- simply deadwrong?

The particular statement that measuring one variable causes you to lose information about the other is more a consequence of the uncertainty principle, not the UP itself. So it's not an incorrect statement, but it's not really a description of the UP either.

The story behind that is that when you measure a particular quantity for a system in a quantum state, the state of the system becomes an eigenstate of the associated operator, with a definite value of that quantity. For example, if you measure a quantum system's position, then immediately after the measurement is made, the position of the system is fixed at exactly the value you measured, with essentially zero uncertainty. The uncertainty principle then says that the uncertainty in the conjugate variable, momentum, has to be infinite, in such a way that their product is at least $\hbar/2$. Now, if you compute the time evolution of a state with an infinitely wide momentum distribution, you'll find that its position distribution spreads out rapidly. That's the diffraction you observed. So it's not that the diffraction made the momentum undetermined; rather, it was the spread in momentum that caused the diffraction. And the spread in momentum came about because, according to the UP, it's impossible to have a state confined to a small space without giving it a large uncertainty in momentum.

 

[prajor]

That view I would say is wrong. It's not actually true that you have to let some time pass to determine momentum - at least, not in theory. At any single instant of time, any physical system is in some state.

In theory, yes we would have the equation of motion and we could calculate position and momentum my using the equation of motion. But HUP talks more about uncertainilty measurement rather than calculation.

so even if we have two sets of apparatus one to measure position and one momentum (completely synchronized), once you measure a position P, you would need a time $$\Delta$$t to measure momentum. so we would not know the position and momentum at the same time.

 

[diazona]

Actually the HUP applies to calculations of position and momentum as well. It's a mathematical identity.

Also, in the part of my post you quoted, I wasn't talking about measuring position, then letting some time pass, then measuring momentum. I was talking about measuring momentum on a system that had not previously been measured at all since it was put in a particular known quantum state.

 

[comote]

I like to view the uncertainty principle as a consequence of the wave nature of QM. The wave function in position space and the wave function in momentum space are Fourier transforms of each other. The more you localize one of the wave functions, the more the Fourier transform smears out. It is actually a result common to any form of "radar".

As for using multiple position measurements to measure momentum, I was of the understanding that one uses the particle nature of light to measure an objects position via diffraction and one uses the wave nature of light to measure the momentum via the Doppler affect. A shorter wavelength will do better at detecting the position of the particle and a longer wavelength will do better at detecting the momentum. Granted my information may be outdated a bit.

 

[Fra]

the right hand side of the inequality is dependent on the confidence measure/certainty you choose, in other words 95%, 99% certainty, ... So is this true? This would make the general statement $$\Delta x \Delta p = O(h)$$.

He probably means that if with $$\Delta x$$ you mean a general confidence INTERVAL, then it's correct that it depends on the confidence LEVEL.

It's just that the custom way of writing HUP refers to the confidence interval defined by 1 standard deviation; which by definition means the confidence level is 68.2%. It would be equallty possible use another "convention" to user a higher confidence level. As long as one knows what 1 standard deviation means it's just a convention.

/Fredrik

 

[Fra]

The UP often gets argued as being a physical consequence, graphically: for knowing its momentum, you have to let some time pass, making its position at that moment less well-defined due to movement

Perhaps we shouldn't confuse time with position, but I think I think I see what you mean.

There is an analogy with time and energy. Anyone that is working with online analysis during dataacquisition and does FFT on sliding windows, then the more accurate you what you know the frequency, the bigger window do you need - the more uncertainty in time for agiven sampling rate. There is also an uncertainty between frequency and time domain.

With x-p it's similar, but it's not that we need a certain amount of "time", we rather need a certain amount of x-data to be able to computer the p-data with a given accuracy. How much "time" it takes to acquire a certain amount of x-data is somehow a more specific.

But with regards to the immediate information STATE; at a given instant of time, then the data we talk about is historic data, each information state has a history and past. So we still have a definite information state at each instant.

/Fredrik

 

[hollowsolid]

In theory, yes we would have the equation of motion and we could calculate position and momentum my using the equation of motion. But HUP talks more about uncertainilty measurement rather than calculation.

so even if we have two sets of apparatus one to measure position and one momentum (completely synchronized), once you measure a position P, you would need a time $$\Delta$$t to measure momentum. so we would not know the position and momentum at the same time.

I agree. To me the essence of the HUP is the sloppiness we have in defining *simultaneity*.

We tend to use the concepts of synchronisation, "same instant" and simultaneity without
thinking that at the QM level it may not exist (or even at the macro level where it's less
critical). i.e. for me the crux of non-commuting variables is that their order of measurement
defines the inequality.

Order implies time and hence the uncertainty lies in the impossibility of 2 events (observations) occupying *exactly* the same time window.

Time spent resolving the x position degrades the resolution of momentum
since at the fundamental level one must precede the other- however slightly.

Of course we can resolve x to a high degree and lose resolution on mv or vice versa
but in these cases the reality that all measurements are sequential is even more
apparent.

 

[DrChinese]

Order implies time and hence the uncertainty lies in the impossibility of 2 events (observations) occupying *exactly* the same time window.

Of course with entangled particle pairs, you can see that the ordering issues really don't change anything. The limit is not because you can't do 2 measurements at exactly the same time.

 

[nonequilibrium]

Hm, thank you all for posting. It's hard to keep track; maybe I should wait until I have my first real class of QM (instead of an introduction to modern physics).

 

[hollowsolid]

Of course with entangled particle pairs, you can see that the ordering issues really don't change anything. The limit is not because you can't do 2 measurements at exactly the same time.

Could you explain the experimental setup involved here? That would help my understanding a lot.

cheers

 

[nonequilibrium]

Hello,

So I've been trying to understand this a little bit more. Thanks for all the posts.

If it is statistical, what does that actually mean? Okay it means the UP is talking about standard deviations, alright, but where does the UP itself come from? Does it follow from some physical postulate?

And for the people interpreting it as a result of the wave-nature of matter: is this compatible with the statistical view?
PS: doesn't an electron, for example, stop being a wavy mess when we "measure" it (whatever that constitutes), yet the UP always persists(?)

 

[Fra]

If it is statistical, what does that actually mean? Okay it means the UP is talking about standard deviations, alright, but where does the UP itself come from? Does it follow from some physical postulate?

Yes it follows from postulates or definitions (depending on how you see it). You can introduce it in different ways, either by postulating commutator relations or by postulating or defining the QM operators themselves from which HUP follows. The mainstream theories so far does not provide any deeper theoretical justification of WHY this is the way it is.

If you start to ask these questions one also has to ask what IS the wavefunction really? In what sense does an information STATE, encode "statistics"? Should be think of this statistics as actual statistics or some imagine ensemble-style statistics? How is that justified given that the information state evolves in time?

In Classical mechanics the DEFINITION of momentum is as we know DIFFERENT. So technically the DEFINITION of terms is totally different in QM, the postulate or conjecture is of course that there is a correspondence in the classical limit. And this conjecture has proved fruitful of course, but still there is no deeper theoretical justification of this in the mainstream. Ie. anyone studying QM in some university, never gets an answer to these questions.

But of course one should not be seduced by magic definitions or postulates and think that there are deductive paths to these things, as the question is still of course why it's exactly THESE postulates that lead to abstractions suitable for formulating successful theories? :)

/Fredrik

 

[diazona]

If it is statistical, what does that actually mean? Okay it means the UP is talking about standard deviations, alright, but where does the UP itself come from? Does it follow from some physical postulate?

It follows from a mathematical identity. (I guess you could say it also follows from the physical postulate that particles are represented by wavefunctions, but beyond that, it's all math.) See Wikipedia for a brief derivation.