I Uncertainty Principle.... Intent Behind It?

[weezy]

The deep answer is symmetry.

See Chapter 3 Ballentine.

And no it can't be explained simply - the proof is deep and tricky - but the result is - well - beauty incarnate.

Thanks
Bill

I will be very grateful if you are able to provide me with a link for this.

 

[bhobba]

I will be very grateful if you are able to provide me with a link for this.

Sorry - you need to get the textbook:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

It derives the actual operators from which the commutation relations trivially follows.

Thanks
Bill

 

[smilodont]

If you measure any fundamental observable very precisely - say position - any non-commuting partner observable moves into a superposition of states (momentum, for example). When you later measure that, you will find that the new outcome (for momentum) is random and uncorrelated to any prior measurement of that observable.

Keep in mind that for all practical purposes, particles do not have simultaneous (well-defined) values for both position and momentum. Experiments on entangled particle pairs demonstrate this very convincingly.

You said, 'When you later measure that, you will find that the new outcome (for momentum) is random and uncorrelated to any prior measurement of that observable.'
Why is this? I get that it is not a matter of us having messed anything up due to the act of measurement. Is it a matter of inconsistent timing? I.e., you measured two things at different times and they MAY have changed between times? Not sure, so I ask.

 

[bhobba]

Why is this?

Its because of the commutation relationship between position and momentum operators:
http://physics.stackexchange.com/questions/10362/how-does-non-commutativity-lead-to-uncertainty

As I mentioned the deep reason they have the form they do is symmetry, but, although very beautiful, its an advanced topic.

Another reason, although not quite as strikingly deep and beautiful, is the connection between commutation and Poisson brackets:
http://bolvan.ph.utexas.edu/~vadim/Classes/2017s/brackets.pdf

Thanks
Bill

 

[weezy]

Its because of the commutation relationship between position and momentum operators:
http://physics.stackexchange.com/questions/10362/how-does-non-commutativity-lead-to-uncertainty

http://physics.stackexchange.com/questions/10362/how-does-non-commutativity-lead-to-uncertainty

http://physics.stackexchange.com/questions/10362/how-does-non-commutativity-lead-to-uncertainty

The first answer was brilliant. Although I have a doubt with the statement that not many operators exist that can transform a Ket to a null vector. Is it really so?

 

[bhobba]

The first answer was brilliant. Although I have a doubt with the statement that not many operators exist that can transform a Ket to a null vector. Is it really so?

Yes its so. It means linear operators with an inverse. If AX = 0 A(-1)AX = 0 ie X = 0.

The usual operators in QM are invertable because they are considered diagonalizable ie of the form A = ∑ yi |xi><xi| Let A' = ∑ 1/yi |xi><xi|. A'A = I.

If the above is goobly gook you need to study linear algebra in bra ket notation:
http://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf

Thanks
Bill

 

[weezy]

Yes its so. It means linear operators with an inverse. If AX = 0 A(-1)AX = 0 ie X = 0.

The usual operators in QM are invertable because they are considered diagonalizable ie of the form A = ∑ yi |xi><xi| Let A' = ∑ 1/yi |xi><xi|. A'A = I.

If the above is goobly gook you need to study linear algebra in bra ket notation:
http://quantum.phys.cmu.edu/CQT/chaps/cqt03.pdf

Thanks
Bill

Thank you for clearing my doubts.

 

[smilodont]

Then why do most introductory textbooks explain HUP using the wave-packet picture? I am referring to Arthur Beiser's modern physics where he shows how the uncertainty relations come about from wave-particle picture of quantum objects and how Fourier transforms relate position and momentum. I am also familiar with Griffiths' explanation where he uses standard deviations to derive and commutator relations to arrive at HUP. But gives no explanation why the operators don't commute.

You said, 'The reason for the HUP is simply the math of non-commuting operators.' By this, do you mean that the way this math works forces you to be 'uncertain'? If so, this seems like an artifact of the math, rather than a result of correct modelling. Or, to put it another way, if we modeled it some other way, it wouldn't be 'uncertain'? Correct?

 

[Nugatory]

You said, 'The reason for the HUP is simply the math of non-commuting operators.' By this, do you mean that the way this math works forces you to be 'uncertain'? If so, this seems like an artifact of the math, rather than a result of correct modelling. Or, to put it another way, if we modeled it some other way, it wouldn't be 'uncertain'? Correct?

Yes, but only helpful if that hypothetical other model also produces predictions that match the results of the countless experiments that agree with the model that we're using - and such a thing doesn't seem to be on offer anywhere. We don't use the mathematical structure of quantum mechanics because we revel in perverse and counterintuitive results, we use it because out of all the countless mathematical models out there, it's the one that best describes the universe we live in.

That doesn't mean that you have to like it, of course. Einstein went to his grave convinced that there had to be something better - and didn't live to see the wave of experimental results that have proven that any alternative model would have to retain all the features that he found most distasteful in QM.

 

[smilodont]

Yes, but only helpful if that hypothetical other model also produces predictions that match the results of the countless experiments that agree with the model that we're using - and such a thing doesn't seem to be on offer anywhere. We don't use the mathematical structure of quantum mechanics because we revel in perverse and counterintuitive results, we use it because out of all the countless mathematical models out there, it's the one that best describes the universe we live in.

That doesn't mean that you have to like it, of course. Einstein went to his grave convinced that there had to be something better - and didn't live to see the wave of experimental results that have proven that any alternative model would have to retain all the features that he found most distasteful in QM.

Then Einstein was right, if we notice the uncertainty caused by the mathematics. This is a sign of something that is an issue. No insult meant here to anyone.

But, I appreciate the answer. I have gotten vague answers before and this helps. Thanks.

 

[Nugatory]

Then Einstein was right, if we notice the uncertainty caused by the mathematics. This is a sign of something that is an issue.

The uncertainty is not caused by the mathematics, it is described by the mathematics. The only question is whether that description is accurate, and all the evidence that we have says that it is.

We can speculate about the possibility that there is some other mathematical model that would work as well as QM and doesn't have the uncertainty principle baked into its structure, but:
1) No one has been able to find one.
2) There are many experiments (most of them involving pairs of non-commuting observables other than position and momentum) that appear impossible to explain by any such hypothetical alternative mathematical model.
3) The only reason for thinking that such an alternative model must exist is that you find the current model so distasteful that you can't accept it, that there must be a better answer. Certainly you have every right to feel that way... But there's no reason to think that universe cares about whether we like its rules.

It's worth noting that once you learn the mathematical justification for the uncertainty principle, you'll see that it actually has a deep and subtle beauty of its own. It's way more elegant and compelling than the obsolete hand-waving explanation of how you have to disturb the particle and change its momentum to find its position. So as you learn more of the real story you will find that the uncertainty principle is less distasteful than it seems at first sight.

 

[bhobba]

The uncertainty is not caused by the mathematics, it is described by the mathematics.

Nothing is caused by math - only described by it.

Its simply a language that has proven particularly suited to physics and related areas.

Sometimes it can be translated to English and explained that way, but in QM mostly it cant. Unfortunately this is one of that mostly.

Thanks
Bill

 

[hilbert2]

Saying that you have all three components of a particle's angular momentum is equivalent to saying that you have a wave function that is an eigenfunction of $L_x$, $L_y$, and $L_z$. That's like saying that you have a triangle with four sides, or the factors of a prime number, or an odd number that is divisible by 2... There is no such thing.

Except for the particular case where all the components of angular momentum are zero.

 

[Zafa Pi]

To me, the uncertainty principle seems to reference our (relatively) poorly controlled methods to measure a particle's momentum and position rather than being some special quantum phenomenon. Is this how it was intended?

This is emphatically not the content of the uncertainty principle. The measurement of of the position of an electron in a particular state |ψ⟩ is the result of a random variable. If we let the standard deviation of that random variable be ∆position(|ψ⟩) and then do the same for the measurement of the momentum we get:
∆position(|ψ⟩)•∆momentum(|ψ⟩) ≥ k > 0 independent of |ψ⟩. That's the H.U.P.
So if the state = wave function |ψ⟩ approximates a delta function at a particular value then ∆position(|ψ⟩) will be small, and thus ∆momentum(|ψ⟩) will be large.

 

[phinds]

Nothing is caused by math - only described by it.

Well, I wouldn't go THAT far. I'm pretty sure it has caused several headaches for me over the years.

 

[lox_and_whiskey]

My understanding of the principle is that we can never be sure about anything at all. or for that matter truth can never be known to the human mind. this principle may have philosophical implications

A more correct reading would be that infinite accuracy in measurement or prediction is not possible. But people don't have goals like that in ordinary life. The human race wants to build taller buildings and faster computers and the like. There are infinite ways to do these things; Physics simply reduces them from infinity to a finite set.

 

[bhobba]

A more correct reading would be that infinite accuracy in measurement or prediction is not possible.

As explained previously it places no restriction on accuracy. And yes it is a general characteristic of QM that you can't predict the outcome of observations.

Thanks
Bill

 

[lox_and_whiskey]

As explained previously it places no restriction on accuracy. And yes it is a general characteristic of QM that you can't predict the outcome of observations.

Thanks
Bill

Oh. Not accuracy then.

 

[Zafa Pi]

A more correct reading would be that infinite accuracy in measurement or prediction is not possible. But people don't have goals like that in ordinary life. The human race wants to build taller buildings and faster computers and the like. There are infinite ways to do these things; Physics simply reduces them from infinity to a finite set.

NO! That is not what is going on with the H.U.P. bhobba (post #67) is correct, in fact measurement accuracy is not the issue. Here is a simple concrete example:
Let |0⟩ = the vector [1,0] (Q-computing notation), now measure that with the observable X and you get +1 with probability ½ and -1 with probability ½. The standard deviation of the measurements is 1. Accuracy is not involved. Now we go into a q-optics lab and send horizontally polarized photons into a polarization analyzer set at 45º and observe the exit ports labeled +1 and -1. We will confirm the QM prediction with our measurements/observations being 100% accurate. Random results.
In fact, variance(Zmeasurement|ψ⟩) + variance(Xmeasurement|ψ⟩) = 1 quite analogous to the H.U.P. Notice, if |ψ⟩ is near |0⟩ then variance(Zmeasurement|ψ⟩) is near 0 (small) while variance(Xmeasurement|ψ⟩) is near 1 (large).

 

[phinds]

Oh. Not accuracy then.

Right. Accuracy and precision are completely possible, repeatability is not. That's basically what the HUP says. If you set up quantum experiments EXACTLY the same every time, you get different results. That's the difference between the quantum world (the real world we live in) and the classical world which turns out to be only an ideal (although it works really well at the macro level)

 

[lox_and_whiskey]

Ahh. So no matter how much you control the setup for an experiment, every property will deviate at least a little between runs. And the closer you fix one property to a $\sigma = 0$ deviation, the more the other will vary, if the product of the deviations is sufficiently small? (for pairs like position and momentum)

 

[Zafa Pi]

Ahh. So no matter how much you control the setup for an experiment, every property will deviate at least a little between runs. And the closer you fix one property to a $\sigma = 0$ deviation, the more the other will vary, if the product of the deviations is sufficiently small? (for pairs like position and momentum)

You can't make $\sigma = 0$ for either position or momentum measurements since their eigenstates are not in the Hilbert space.