Heisenberg's uncertainty principle

[DannyTr]

Instead of just taking one measurement of the particle, you take a 2nd measurement in addition thereby gathering more information about the particle then the uncertainty principle allows?It would be possible to extend out to an arbitrary number of follow-on measurements thereby measuring position and momentum to arbitrary precision?

 

[PeroK]

Instead of just taking one measurement of the particle, you take a 2nd measurement in addition thereby gathering more information about the particle then the uncertainty principle allows?It would be possible to extend out to an arbitrary number of follow-on measurements thereby measuring position and momentum to arbitrary precision?

The HUP (Heisenberg Uncertainty Principle) is a statistical law. In the equation:

$\sigma_x \sigma_p \ge \frac{\hbar}{2}$

The $\sigma$ represents the standard deviation of a set of measurements on a large set of identically prepared particles. If the particle is in a state where there is little variation in the expected value of position measurements, then the HUP says that there must be a large variation in the expected value of momentum measurements, and vice versa.

If you measure the position of a particle very accurately, there is nothing to stop you immediately measuring its momentum very accurately. But, what the HUP says, is that if you repeated this experiment many times from the same starting state, then the range of momentum measurements you get would be large. There is the difference between an accurate or precise measurement of momentum and a large variation in the (precise) expected momentum measurements.

The HUP, therefore, says nothing about how precisely you can measure position and momentum, but does say something about the range of expected values you will get.

 

[DannyTr]

The HUP (Heisenberg Uncertainty Principle) is a statistical law. In the equation:

$\sigma_x \sigma_p \ge \frac{\hbar}{2}$

The $\sigma$ represents the standard deviation of a set of measurements on a large set of identically prepared particles. If the particle is in a state where there is little variation in the expected value of position measurements, then the HUP says that there must be a large variation in the expected value of momentum measurements, and vice versa.

If you measure the position of a particle very accurately, there is nothing to stop you immediately measuring its momentum very accurately. But, what the HUP says, is that if you repeated this experiment many times from the same starting state, then the range of momentum measurements you get would be large. There is the difference between an accurate or precise measurement of momentum and a large variation in the (precise) expected momentum measurements.

The HUP, therefore, says nothing about how precisely you can measure position and momentum, but does say something about the range of expected values you will get.

- I mean repeat the measurement many times for a SINGLE PARTICLE:
- The particle is measured in instrument 1.
- The particle deflects and is measured by instrument 2.
- And so on until a to arbitrary precision for both the original position and momentum are in obtained via trigonometry

 

[PeroK]

- I mean repeat the measurement many times for a SINGLE PARTICLE:
- The particle is measured in instrument 1.
- The particle deflects and is measured by instrument 2.
- And so on until a to arbitrary precision for both the original position and momentum are in obtained via trigonometry

The position and momentum of a particle are, however, affected by the measurement process. So, you only get one chance for the particle in its original state. As soon as you make one measurement, it is no longer in its original state.

Also, to reemphasise, the HUP is a statistical law, so does not directly govern repeated measurements on a single particle.

 

[DannyTr]

The position and momentum of a particle are, however, affected by the measurement process. So, you only get one chance for the particle in its original state. As soon as you make one measurement, it is no longer in its original state.

Also, to repeat, the HUP is a statistical law, so does not directly govern repeated measurements on a single particle.

- I know you only get one initial measurement but taking a 2nd measurement after the particle deflects from the first gives you the angle of deflection which is going to narrow uncertainty over position and momentum

 

[PeroK]

- I know you only get one initial measurement but taking a 2nd measurement after the particle deflects from the first gives you the angle of deflection which is going to narrow uncertainty over position and momentum

... you're not listening.

The HUP is a statistical law. It says nothing about repeated measurements on a single particle.
The HUP is a statistical law. It says nothing about repeated measurements on a single particle.
The HUP is a statistical law. It says nothing about repeated measurements on a single particle.
...

Your example is like tossing a coin and, because it doesn't come up half-heads and half-tails, declaring that the probability theory that predicts 50% heads and 50% tails is violated.

 

[PeroK]

- I know you only get one initial measurement but taking a 2nd measurement after the particle deflects from the first gives you the angle of deflection which is going to narrow uncertainty over position and momentum

Also, according to QM, a particle does not have a position or momentum. You cannot talk about the original position and momentum. You can only talk about the state of a particle, which implies the probabilities of getting different values of position and momentum if you carry out a measurement.

The uncertainty about a particle, therefore, goes much deeper than simply not knowing its position and momentum. It does not have a position or momentum until you carry out a measurement.

 

[DannyTr]

But I thought people took the HUP as a Fundamental resolution limit?

If it is in fact possible to collect more information simultaneously about a particle than the HUP says is that not something worth knowing?

 

[PeroK]

But I thought people took the HUP as a Fundamental resolution limit?

Yes, many people do. And they are all wrong.

If it is in fact possible to collect more information simultaneously about a particle than the HUP says is that not something worth knowing?

The state of a particle tells you everything (simultaneously) about the position and momentum of a particle. But, what it tells you is the probabilities of getting certain measured values. The HUP is a relation between these probabilities.

In that sense, there is nothing more to know.

 

[DrChinese]

But I thought people took the HUP as a Fundamental resolution limit?

Yes, it is fundamental (although you can also consider it as derived from fundamental considerations as well). And although it is in fact a statistical limit, as already mentioned, there are direct physical issues around this you may not be aware of that apply to a single particle. Specifically: if you know with great certainty observable X on a particle, and then measure NON-COMMUTING observable Y on the same particle with great precision/certainty: its X observable is now equally UNCERTAIN. So it will now have a random value of X observable. What have you really learned? Always just one thing or the other, never both at the same time.

Note that the HUP does not constrain commuting observables.

 

[PeroK]

Yes, it is fundamental (although you can also consider it as derived from fundamental considerations as well). And although it is in fact a statistical limit, as already mentioned, there are direct physical issues around this you may not be aware of that apply to a single particle. Specifically: if you know with great certainty observable X on a particle, and then measure NON-COMMUTING observable Y on the same particle with great precision/certainty: its X observable is now equally UNCERTAIN. So it will now have a random value of X observable. What have you really learned? Always just one thing or the other, never both at the same time.

Note that the HUP does not constrain commuting observables.

The thing to avoid, as I know you know, is to see the HUP as a constraint on the measurement itself. That the HUP prevents a precise measurement in some way. That if you follow a precise measurement of position with a measurement of momentum, then the measurement of momentum is imprecise or constrained in some way. Instead its the measurement value that is "uncertain" in the sense that it can inherently take a wide range of values.

 

[DrChinese]

The thing to avoid, as I know you know, is to see the HUP as a constraint on the measurement itself. That the HUP prevents a precise measurement in some way.

Of course I agree. And hopefully the OP won't read what I said as implying a limitation on the precision of any single measurement. Theoretically, there is nothing particularly limiting us on that score past technology.

What I hope Danny walks away with is the idea that a very certain measurement of one observable (say B) renders its non-commuting sibling observable (say A) completely indefinite. By indefinite I mean: A has no specific value, and if A is subsequently measured, the newly measured value of A would have nothing more than a random relationship to any previously measured value of A.

I would contrast that with even more follow-on measurements of A, all of which could potentially continue to yield the same definite value. Therefore: measurements in and of themselves to not force a quantum observable to change.

 

[bobob]

The thing to avoid, as I know you know, is to see the HUP as a constraint on the measurement itself. That the HUP prevents a precise measurement in some way. That if you follow a precise measurement of position with a measurement of momentum, then the measurement of momentum is imprecise or constrained in some way. Instead its the measurement value that is "uncertain" in the sense that it can inherently take a wide range of values.

No, it's a constraint on what there is to measure. The rules "nature" uses to measure thing are the same as ours.

 

[edmund cavendish]

And the "never both at the same time" does not weaken or invalidate the measurement of either or imbue the meaurer/ observer with some "effect on the observed"

 

[vanhees71]

It's often misunderstood that the Heisenberg uncertainty principle is saying something about the "possiblity to measure" an observable or that it's about the inevitable influence of a measurement, which means that the measured system is interacting with the measurement apparatus, on the measured system. That's wrong, and these questions have to be treated other than with the Heisenberg uncertainty relation. Interestingly Heisenberg himself got his own principle wrong in the first place and was corrected by Bohr. Of course if you have two philosophers at work the subject gets as obscured as possible, and thus the misunderstanding got propagated through the textbook history.

The Heisenberg uncertainty relation, which reads in its most general form
$$\Delta A \Delta B \geq |\langle \mathrm{i} [\hat{A},\hat{B}] \rangle|,$$
where $A$ and $B$ are arbitrary observables, $\Delta A$ and $\Delta B$ the standard deviations when these observables are measured on an ensemble representing an arbitrary quantum state $\hat{\rho}$. All averages, i.e., to evaluate the standard deviations and the expectation value on the right-hand side have to be taken with respect to this prepared state, represented by the Statistical Operator $\hat{\rho}$. Thus the uncertainty relations are a statement about the possibility to prepare a quantum system in any quantum state possible. It doesn't tell you anything about the possibility to measure the observables on an ensemble nor about the influence of the measurement apparatus on the system, which of course cannot be generally stated, but depends on the details of how the measurement device is constructed.

 

[f95toli]

The thing to avoid, as I know you know, is to see the HUP as a constraint on the measurement itself.

I think this is partly because several issues get confused. The HUP certainly has "consequences" if you discuss measurements (which is why we talk about things like quantum limited amplifiers) but generally speaking they are indeed "indirect" and often somewhat difficult to get a firm handle on (they simply "pop up" in the math).

 

[ZapperZ]

- I mean repeat the measurement many times for a SINGLE PARTICLE:
- The particle is measured in instrument 1.
- The particle deflects and is measured by instrument 2.
- And so on until a to arbitrary precision for both the original position and momentum are in obtained via trigonometry

You need to go back to the beginning and figure out WHAT are you trying to measure. It is not just any old position measurement or any old momentum measurement.

For example, let's say I want to measure the momentum of an electron that was just liberated off a surface. If I make a "position" measurement first, I will have disrupted the information about its original momentum. Now, there's nothing to stop me from measuring its momentum later, but what is that significance of THAT momentum based on what I initially wanted? In other words, what does the momentum that I measure have anything to do with the momentum of the electron JUST after it was liberated?

You can make as many measurements as you want, but each measurement disrupts the original information about the property that you want to measure. You are losing "coherence" on the original properties of that entity. After one, or even two measurements, what you are measuring will have little resemblance left to what it was originally.

BTW, you need to look at the many Insights and FAQs on what the HUP really is. The measurement of ONE parameter (say position) has no statistical spread, and thus, it isn't what the HUP is saying. So my impression here is that you do not yet have a proper grasp on what the HUP is.

Zz.

 

[vela]

Thus the uncertainty relations are a statement about the possibility to prepare a quantum system in any quantum state possible.

I think this is the best way to explain the nature of the uncertainty principle. I think going on about ensembles obscures this fact and just confuses beginners.

The OP is assuming that a particle simultaneously has a well-defined location and well-defined momentum, so it's just a matter of being really clever about how to measure those quantities. Once you understand that the uncertainty principle says that such a quantum state is impossible, you have to accept that no amount of cleverness is going to allow you to measure quantities that don't really exist.

 

[vanhees71]

Well, I've no other way to understand the practical meaning of probabilities than the good old frequentist interpretation, i.e., to observe probabilities I have to prepare somehow an ensemble and measure the observable(s) I want to have statistical knowledge about, and that's how measurements in the lab are done.

Then it's wrong to say, you cannot measure and observable, because the system you measure is not prepared in an eigenstate of the corresponding operator. You can always measure any observable you can define on a system (at least in principle). The only thing QT tells you is that the observable doesn't take a determined value, if the system isn't prepared such that this observable has a determined value.

 

[haushofer]

Just to add my 2 cents: I think vanhees' mentioning of an ensemble is very important, at least, it was what confused me in the beginning. Instead of thinking about measurements being performed on 1 system, you should think about measuring on elements of an ensemble of "identically prepared systems", one at a time. Only then you can speak about a population and corresponding standard deviation.