I Can We Accurately Measure Momentum in the Lab?

[dsaun777]

In the lab, how accurately can we measure momentum? What is the max value of the uncertainty in position as the uncertainty in momentum approaches zero? Or vice versa. What experiments do these types of measurements?

 

[PeroK]

In the lab, how accurately can we measure momentum? What is the max value of the uncertainty in position as the uncertainty in momentum approaches zero? Or vice versa. What experiments do these types of measurements?

The Heisenberg Uncertainty Principle is not about the accuracy of measurements. It's a statistical law about the variance of measurements on an ensemble of identically prepared systems.

 

[PeroK]

That said, the single slit experiment demonstrates the HUP to some extent. As the slit gets narrower, the range of lateral momentum gets greater. But, also, less light gets through, so the pattern gets fainter.

I'm not sure what is the physical limit on the width of the slit. You could Google that.

 

[dsaun777]

The Heisenberg Uncertainty Principle is not about the accuracy of measurements. It's a statistical law about the variance of measurements on an ensemble of identically prepared systems.

But the hup puts a limit on measurements. Forget about mathematical derivation of hup, I am interested in how the hup influence measurements.

 

[PeterDonis]

the hup puts a limit on measurements

It does not put any limit on the accuracy of individual measurements. It only puts limits on the statistics of ensembles of measurements of non-commuting observables.

 

[Nugatory]

But the hup puts a limit on measurements.

That is a common misconception, one that made it into the popular consciousness almost a century ago and has persisted as sort of urban legend ever since.

What the uncertainty principle does say is something along the lines of...
The state in which a position measurement will give us an exact infinite-precision result X is also a state in which we cannot exactly predict the result of an exact infinite-precision measurement of the momentum P.

We test this proposition by repeating the same experiment: initialize the system in the state such that if we measure the position we will get X; and then measure the momentum instead. Do this over and over again and we will find that even though the initial state is the same every time, and even though we are doing an exact measurement of P, we get different results for P each time. The uncertainty principle describes the statistical spread of these results.
(You should be aware that I have cut some corners, as well as ignoring the mathematical complications that come from position and momentum being continuous variables)

 

[hutchphd]

My two cents:

1. If I wish to measure, say, a momentum in an experiment, at some point I compare it to a `"known" momentum and this can be a very involuted process with experimental uncertainties at each step. These can usuually be ameliorated by performing repeated identical experiments.
2. The Heisenberg Uncertainty Principle is fundamentally a statement about information and how it is internally shared (in the mind/hand of god if you wish). Apparently one can only arbitrarilly specify a limited number of attributes self-consistently. No clever technique will make it better.
3. Honestly how could the universe be different?

These two uncertainties are often compounded but they are fundamentally distinct.

 

[dsaun777]

So the single measured momentum in a single experiment is the exact momentum for that measurement. It's only when you try and reproduce that measurement that the uncertainty raises?

 

[PeroK]

So the single measured momentum in a single experiment is the exact momentum for that measurement. It's only when you try and reproduce that measurement that the uncertainty raises?

The uncertainty in any particular measurement depends on the measurement process. Not on the HUP. The theoretical, minimum variance (to use the precise statistical term) over a number of measurements on identical systems is determined by the HUP.

In particular the HUP specifies a lower bound for the product of the variance in momentum and position in any system.
$$\sigma_x \sigma_p \ge \frac \hbar 2$$

 

[dsaun777]

The uncertainty in any particular measurement depends on the measurement process. Not on the HUP. The theoretical, minimum variance (to use the precise statistical term) over a number of measurements on identical systems is determined by the HUP.

In particular the HUP specifies a lower bound for the product of the variance in momentum and position in any system.
$$\sigma_x \sigma_p \ge \frac \hbar 2$$

How do you define a measured momentum in physics? What device is used? how is it we have such exact values of certain particle momenta like photons and electrons? For instance, we want to know the wavelength of light from the sun. how close is our measurement of such wavelengths to the actual wavelength?

 

[PeroK]

How do you define a measured momentum in physics? What device is used? how is it we have such exact values of certain particle momenta like photons and electrons?

try this:

https://arxiv.org/abs/2302.12303

For instance, we want to know the wavelength of light from the sun. how close is our measurement of such wavelengths to the actual wavelength?

There is no "actual" wavelength. There is only the measured wavelength. That's quite fundamental to QM.

 

[dsaun777]

try this:

https://arxiv.org/abs/2302.12303

There is no "actual" wavelength. There is only the measured wavelength. That's quite fundamental to QM.

Thank you for the paper. It appears that the time of flight method of slit experiments that utilize transverse momentum is the most common method for measuring momentum. And there is no discussed limit on how accurately you can measure the momentum from a "single shot." It is only when you introduce the second, third, etc do you get the generated statistical ensemble of measurement results that shows up in the HUP. The atom trap method was interesting I haven't heard of that yet, but a similar principle to the slit measurement.

 

[DrChinese]

But the hup puts a limit on measurements. Forget about mathematical derivation of hup, I am interested in how the hup influence measurements.

The HUP only specifies measurement limits on PAIRS of measurements on NON-commuting observables. Using your example of momentum and position:

a) You can perform a very accurate measurement of momentum on a particle, and suppose the observed value is P. A repeated measurement of the momentum will also yield a value of P (or very close to it, assuming that the original measurement did not otherwise disturb the particle).

b) You can then perform a very accurate measurement of position on that same particle, and suppose the value is Q. This has the effect of changing the future observed momentum of the particle regardless of how the position measurement is carried out. You could say that the P momentum previously observed is rendered meaningless, and is given a new value that is randomly assigned.

c) You can then perform another very accurate measurement of momentum on a particle, and suppose the value is P'. The HUP essentially says that P and P' will no longer bear any relationship to each other, at least no closer than the statistical limits which are related to the mathematical precision of the HUP.

On the other hand: for observables which COMMUTE: there is no HUP restriction or limitation. Momentum and spin commute. So a spin measurement does not intrinsically invalidate a previous momentum measurement on a particle. Repeated measurements of spin and momentum will yield a constant value (assuming the method of measurement does not otherwise change these values).

Please note that my example uses phraseology that will necessarily raise objections from some folks. My objective is to emphasize the basics of how the HUP works as it pertains to your question in the OP. Repeated measurements of the same observable is not limited by the HUP itself - it would only be limited by the nature and accuracy of the experimental method.

 

[dsaun777]

Ultimately, the HUP is just a mathematical result that is derived from the Fourier transform from wave analysis and should not have any meaning in a single measurement experiment.

 

[phinds]

So the single measured momentum in a single experiment is the exact momentum for that measurement. It's only when you try and reproduce that measurement that the uncertainty raises?

Yes, assuming you have prepared the position exactly the same. That's what @PeroK was talking about when he said

The Heisenberg Uncertainty Principle is not about the accuracy of measurements. It's a statistical law about the variance of measurements on an ensemble of identically prepared systems.

 

[Nugatory]

Ultimately, the HUP is just a mathematical result that is derived from the Fourier transform from wave analysis and should not have any meaning in a single measurement experiment.

It might best to think of it as a limitation on state preparation, one that follows from the non-commutation of some pairs of observables.

 

[vanhees71]

But the hup puts a limit on measurements. Forget about mathematical derivation of hup, I am interested in how the hup influence measurements.

The HUP is often misinterpreted, which has historical reasons. The funny point is that Heisenberg published his first paper on the HUP without discussing its contents with Bohr, and there he interpreted it as if it would mean that you couldn't measure accurately momentum without disturbing the system in such a way that momentum gets more uncertain and vice versa. This is, however, not what the HUP, derived from QM, is saying, and that was corrected by Bohr. Unfortunately the first interpretation somehow stuck within the literature.

The point is, as was pointed out before in this thread, that the usual HUP,
$$\Delta A \Delta B \geq \frac{1}{2} |\langle \mathrm{i} [\hat{A},\hat{B}] \rangle|,$$
is about the statistical properties of measurement results when measuring accurate two observables $A$ and $B$ on ensembles of equally prepared quantum systems. Indeed $\Delta A$ and $\Delta B$ are standard deviations. The standard HUP thus refers not to measurements but to the preparability of the system in a way that the one or the other observable are accurately determined.

In the case of position and momentum it's saying
$$\Delta x_j \Delta p_j \geq \frac{\hbar}{2},$$
i.e., the right-hand side is independent of the prepared quantum state, and that makes this case easy to interpet: If you prepare a particle such that the $j$-component of its position vector, $x_j$, is pretty well determined ($\Delta x_j$ small), then necessarily the $j$-component of its momentum vector $p_j$ cannot be so well prepared ($\Delta p_j$ gets necessarily large).

As any probabilistic statement, from a practical point of view, also the HUP refers to ensembles of equally prepared particles. The preparation in the corresponding state has nothing to do with the possibility to measure either $x_j$ or $p_j$ accurately. Of course such measurements usually need different setups of measurements. You can measure $x_j$ with one setup very accurately on many equally prepared particles. To check the uncertainty relation, the position resolution of the appartus must be much better than $\Delta x_j$, such that you get random positions for any single measurement that you can then analyze statistically and determine $\Delta x_j$. The same you can do for $p_j$. Again your momentum resolution must be much better than $\Delta p_j$ to get significant statistics do determine $\Delta p_j$ in the statistical measurment analysis.

Of course, it's also true that any measurement has an unavoidable influence on the measured system, i.e., indeed there is some disturbance of the system involved in any measurement, and also this can be described by quantum theory, but it's not described by the usual HUP. Rather you have to make a detailed analysis of the specific experiment. Then you get also measurement-disturbance uncertainty relations, but this is not the usual HUP, which is pretty easily derived in any textbook on QM.

A very concise discussion about the important difference between the usual HUP and disturbance through measurement, see

https://arxiv.org/abs/quant-ph/0510083
https://doi.org/10.1088/1464-4266/7/12/033

 

[f95toli]

While the above is correct; you also need to remember that what we often mean by a "single" measurement is actually a statistical process. For example, the HUP does set the lower limit for how much noise a microwave amplifier will add when used in a measurement. Hence, this noise needs to be taken into account if you want to know how precisely you can measure a microwave signal.

This does not contradict anything that was written above; since each single "measurement" will still yield a single well-defined value; the concept of a "noisy measurement" (or noise in general) only make sense if a measurement is actually made up of a (often vary large) number of individual measurements; but as it happens this is how nearly all measurements are done in practice.

This is why there is a huge interest in developing methods for "beating the standard quantum limit"
(squeezing etc),

 

[vanhees71]

In which sense tells you the usual HUP, how much noise a microwave amplifier will add when used in a measurement? I don't see this at all. Do you have a reference for that claim? I think that's more an example of a "noise-deturbance uncertainty relation", which has to be determined by calculating the time evolution of the state under consideration of this amplifier in its interaction with the measured system.

 

[f95toli]

See e.g., section 4 in

Roy, Ananda, and Michel Devoret. "Introduction to parametric amplification of quantum signals with Josephson circuits." Comptes Rendus Physique 17.7 (2016): 740-755.

Also on the arXIv

https://arxiv.org/abs/1605.00539

From the paper:

More practically, the extra half-photon of noise can also be seen as a consequence of the Heisenberg Uncertainty Principle. A phase preserving amplifier processes equally both quadratures, which in quantum mechanics are non-commuting observables. Since the process of amplification is equivalent to measurement, the extra noise forbids that both quadratures are known precisely simultaneously ,in accordance with the central principle of quantum mechanics. An amplifier functioning in this Heisenberg regime where the efficiency of the amplification process is only limited by irreducible quantum fluctuations is said to be quantum-limited.

This is how most people working with these amplifiers (which sort of includes me) think about the "need" for the extra half photon of noise. I believe this is very similar to how people approach similar question in optics where mode squeezing is used to improve read-out performance for some detectors.

 

[vanhees71]

Yes, and as far as I understood from just quick glancing over the paper, it's a nice example for, how such "noise-disturbance relations" for measurements are derived from the dynamics under influence of the measurement device (in this case the amplifier).