Measuring Particle Momentum: Angles without Position?

[LarryS]

Most of the literature that I have read on non-relativistic wave mechanics discusses experiments that measure a particle’s position (double-slit, etc.). But I have found almost no examples of experiments for measuring a particle’s momentum. In general, I know that one measures the momentum of a charged particle by measuring its angle of deflection while passing through a magnetic field. But how does one determine such an angle without knowing the particle’s position at the same time? Please clarify. Thanks in advance.

 

[Fredrik]

There are several other threads about this. This is the most recent one. I don't know if there's a really satisfying answer in any of them. If there is, I haven't seen one. (But then I haven't read many of them either). My thoughts on this are similar to what SpectraCat said in the thread I linked to. I'll even go one step further and say that every measurement only produces a signal that informs us that some interaction has taken place.

I'd be interested in a discussion specifically about particle tracks in a bubble chamber. The path of a charged particle is curved, and the momentum can be calculated from the curvature. Each bubble reflects light differently than the rest of the liquid. This light is the "signal" that tells us that an interaction has taken place in the region where the bubble is located. That interaction has to localize the particle to that region, i.e. make its wavefunction close to zero outside the bubble. Then I suppose the wavefunction immediately starts to spread out, only to be squeezed into a more narrow shape again by the interaction that produces the next bubble, and so on.

Now the question is, how is the momentum essentially unchanged through all this? The answer obviously involves "conservation of momentum", but I think we'd have to look more closely at the actual interaction with the liquid to really understand this.

 

[LarryS]

There are several other threads about this. This is the most recent one. I don't know if there's a really satisfying answer in any of them. If there is, I haven't seen one. (But then I haven't read many of them either). My thoughts on this are similar to what SpectraCat said in the thread I linked to. I'll even go one step further and say that every measurement only produces a signal that informs us that some interaction has taken place.

I'd be interested in a discussion specifically about particle tracks in a bubble chamber. The path of a charged particle is curved, and the momentum can be calculated from the curvature. Each bubble reflects light differently than the rest of the liquid. This light is the "signal" that tells us that an interaction has taken place in the region where the bubble is located. That interaction has to localize the particle to that region, i.e. make its wavefunction close to zero outside the bubble. Then I suppose the wavefunction immediately starts to spread out, only to be squeezed into a more narrow shape again by the interaction that produces the next bubble, and so on.

Now the question is, how is the momentum essentially unchanged through all this? The answer obviously involves "conservation of momentum", but I think we'd have to look more closely at the actual interaction with the liquid to really understand this.

The position-momentum uncertainty relation is taught in undergraduate, introductory QM courses. The mathematics (Fourier transforms, noncommuting operators, ...) is absolutely sound. Nobody questions this. But apparently, there is this wide gap between theory (the math) and reality (the experiments). I find that fascinating.

 

[sweet springs]

Hi, Fredrik
I just drop a comment on bubble chamber or wire net chamber experiment in high energy physics.

I'd be interested in a discussion specifically about particle tracks in a bubble chamber. The path of a charged particle is curved, and the momentum can be calculated from the curvature. Each bubble reflects light differently than the rest of the liquid. This light is the "signal" that tells us that an interaction has taken place in the region where the bubble is located. That interaction has to localize the particle to that region, i.e. make its wavefunction close to zero outside the bubble. Then I suppose the wavefunction immediately starts to spread out, only to be squeezed into a more narrow shape again by the interaction that produces the next bubble, and so on.

Say radius of bubble Δx, momentum error for each positioning is Δp ～ h'/Δx
N times positioning to measure curvature requires accumulated uncertainly NΔp ～ Nh'/Δx approximately.
In case <p> is much greater than Nh'/Δx, the measurement of momentum practically works as it is in high energy physics.

Now the question is, how is the momentum essentially unchanged through all this? The answer obviously involves "conservation of momentum", but I think we'd have to look more closely at the actual interaction with the liquid to really understand this.

So I do not think we require conservation of momentum during the measurement process of N dots.

Further in case <p> and Nh'/Δx are comparative, we can make bubble chamber move at a constant speed so that <p> is much greater than Nh'/Δx in its frame of reference. It seems all right but I appreciate comments of you all.

Regards.

 

[ZapperZ]

Most of the literature that I have read on non-relativistic wave mechanics discusses experiments that measure a particle’s position (double-slit, etc.). But I have found almost no examples of experiments for measuring a particle’s momentum. In general, I know that one measures the momentum of a charged particle by measuring its angle of deflection while passing through a magnetic field. But how does one determine such an angle without knowing the particle’s position at the same time? Please clarify. Thanks in advance.

The "momentum" of a particle is measured in condensed matter physics all the time. This is because one of the most useful way to analyze the property of a material is to look at the structure in reciprocal space or momentum space. Furthermore, the band structure of the material, which is the most useful aspect of the characteristics of a material, is a E vs. k plot, which means that you need to know the momentum (or crystal momentum) of the material.

One of the most common technique to make such a determination is via angle-resolved photoemission spectroscopy (ARPES). My avatar is the raw data from one such technique. The vertical axis is energy while the horizontal axis is the measured momentum of the photoelectrons. This was taken using a Scienta SES 200 hemispherical analyzer. A review of ARPES and the basic physics on how one could resolve what one measures into such E and k values can be found in this http://arxiv.org/PS_cache/cond-mat/pdf/0208/0208504v1.pdf" . Now how, in Eq. 1, 2, and 3, the detected particles can be resolved into its momentum value. The image in Fig. 7 is similar to my avatar. While the image corresponds to where the electron hits the detector, it tells you nothing about the position of the electron while in the material. It tells you, however, the energy and momentum of the electron simultaneously.

Note also that very often, people confused the HUP with the instrument uncertainty of ONE single measurement. The latter is not the HUP and has nothing to do with it (i.e. you can decrease the uncertainty in one as much as you can without affecting the uncertainty in the other). I've discussed this before in other threads using my single-slit diffraction example.

Zz.

 

[sweet springs]

Hi, Zz.
In usual experiments of condensed matter physics, we do not care where electron is in the specimen whose size say L is macro scale. We expect momentum p of the probability density of |ψ(p)|^2 be observed with error of Δp=h'/L=0 from uncertainty relation. However, the spectroscopy process as shown in FIG.6 in the referred paper, we have to make observation of the positions of emitted electrons at the entrance slit and the detector. These positioning harm the nature of the state by uncertainty relation. Δp thus caused by uncertainty relation should be arranged small enough for the purpose of the experiments. Am I right in understanding uncertainty relation?
Regards.

 

[ZapperZ]

Hi, Zz.
In usual experiments of condensed matter physics, we do not care where electron is in the specimen whose size say L is macro scale. We expect momentum p of the probability density of |ψ(p)|^2 be observed with error of Δp=h'/L=0 from uncertainty relation. However, the spectroscopy process as shown in FIG.6 in the referred paper, we have to make observation of the positions of emitted electrons at the entrance slit and the detector. These positioning harm the nature of the state by uncertainty relation. Δp thus caused by uncertainty relation should be arranged small enough for the purpose of the experiments. Am I right in understanding uncertainty relation?
Regards.

The entrance slit isn't small when compared to the deBroglie wavelength of the electrons, i.e. it doesn't cause any diffraction. So the purpose of the slit here isn't as a position measurement. Rather it is to select a range of momentum direction.

Zz.

 

[sweet springs]

Hi, Zz.
Passing the slit moment of electron changes Δｐ～h'/Δx where Δx is the width of the slit. Δx should be large enough so that Δp is much smaller than <p>. On the other hand too large Δx would degrade momentum measurement due to orbit radius ambiguity.
I agree with you that practically we are observing momentum. As a matter of principle, I do not know any momentum measurement process that preserve momentum of the state.
Regards.

 

[ZapperZ]

Hi, Zz.
Passing the slit through moment of electron changes Δｐ～h'/Δx where Δx is the width of the slit. Δx should be large enough so that Δp is much smaller than <p>. On the other hand too large Δx would also degrade momentum measurement due to orbit radius ambiguity.
I agree with you that practically we are observing momentum. As a matter of principle, I do not know any momentum measurement process that preserve momentum of the state.
Regards.

Considering the momentum resolution that we now could get with laser-based ARPES (which could be one of, of not THE highest resolution of momentum anyone could get with any kind of measurement), I think you need to address what you said and show why these experiments continue to report such momentum measurements.

Zz.

 

[sweet springs]

Hi, Zz
I here state again with pleasure and respect that the momentum measurements in these experiments are carefully arranged, precise enough and productive. Regards.

 

[Fredrik]

Say radius of bubble Δx, momentum error for each positioning is Δp ～ h'/Δx
N times positioning to measure curvature requires accumulated uncertainly NΔp ～ Nh'/Δx approximately.
In case <p> is much greater than Nh'/Δx, the measurement of momentum practically works as it is in high energy physics.

I like this explanation. It's easy to understand and it appears to answer my question. I'm feeling a bit too lazy right now to plug in realistic numbers to verify that it all works out, but I'd be surprised if it doesn't.

Further in case <p> and Nh'/Δx are comparative, we can make bubble chamber move at a constant speed so that <p> is much greater than Nh'/Δx in its frame of reference.

I suppose, but it seems difficult to actually do this. I'm getting this image in my head of physicists with 1970's mustaches and hairdos doing experiments with bubble chambers flying around at relativistic speeds.

 

[ZapperZ]

Hi, Zz
I here state again with pleasure and respect that the momentum measurements in these experiments are carefully arranged, precise enough and productive. Regards.

Then I don't understand all of this. ALL experiments are "carefully arranged, precise enough and productive", even the ones that measure "energy" (as in this experiment). So why are we picking on momentum measurement?

I could easily do away with the entrance slit to the analyzer. However, I will then get way too many electrons coming in way too many directions that will cause my data to be filled with noise, the same way you'll get washed out diffraction patterns if you use non-monochromatic light source. In fact, these analyzers come with several different slit sizes where one can, say, use a larger slit if the photoelectrons number is just too small to make a clear signal. So you increase the slit size to admit more electrons, but you compromise on the resolution. But none of these have any connection with the HUP and with the actual ability to measure the momentum of each of the electrons that hit the detector.

Zz.

 

[Bob S]

I'd be interested in a discussion specifically about particle tracks in a bubble chamber. The path of a charged particle is curved, and the momentum can be calculated from the curvature. Each bubble reflects light differently than the rest of the liquid...
Now the question is, how is the momentum essentially unchanged through all this? The answer obviously involves "conservation of momentum", but I think we'd have to look more closely at the actual interaction with the liquid to really understand this.

The bubble track is produced by collisions between the incident particle and atomic electrons in the liquid, causing a transverse momentum and energy transfer to the atomic electrons, described by the Bethe Bloch dE/dx equation. The incident particle energy loss is ~2 MeV per gram/cm² for minimum ionizing particles (~ 4 MeV/gram/cm² for hydrogen). The net particle transverse track deviation is caused by nuclear Coulomb collisions and is called multiple Coulomb scattering.

Bob S

 

[sweet springs]

Hi, Zz
Momentum measurement process of electron spectrometer in FIG.16 of the refferd paper consists of positioning at the entrance slit, bending the orbit by magnetic field, and positioning at the detector.

Q1 Do positionings cause error in measurement of momentum?
Yes, as uncertainty relation says. /No, we can get precise enough information of electron momentum states in the condensed matter by careful arrangement.

Q2 Does bending the orbit harm momentum measurement?
Yes, we change the direction of momentum to be measured by bending. /No, bending the emitted electrons does not affect the momentum states in the condensed matter at all.

I think both the answers of different view points and interests have means.
Regards.

 

[ZapperZ]

Hi, Zz
Momentum measurement process of electron spectrometer in FIG.16 of the refferd paper consists of positioning at the entrance slit, bending the orbit by magnetic field, and positioning at the detector.

Q1 Do positionings cause error in measurement of momentum?
Yes, as uncertainty relation says. /No, we can get precise enough information of electron momentum states in the condensed matter by careful arrangement.

Q2 Does bending the orbit harm momentum measurement?
Yes, we change the direction of momentum to be measured by bending. /No, bending the emitted electrons does not affect the momentum states in the condensed matter at all.

I think both the answers of different view points and interests have means.
Regards.

I have no idea what you just said here.

1. There are NO "electron orbits".

2. The "bending" is done to measure the energy, NOT momentum.

3. In modern electrostatic hemispherical analyzer, even a magnetic field isn't used. A complex series of electrostatic lenses are in place that are carefully calibrated with a standard electron gun.

4. You are obsessed with the "positioning", and I don't understand why. The "positioning" or direction of the analyzer orientation, once again, is simply to collect momenta in a smaller range of directions. After we do that, we then move to another range! This is how we can collect and map the bands in not only the first Brillouin zone, but even the second!

Zz.

 

[sweet springs]

Hi, Zz.
I've misunderstood ARPES. From Ekin=hν-φ-|EB|, (1) in the review article, and p||=h'k||=sqrt(2mEkin)sinθ, (2) in the review, counting photoelectron of Ekin in hemispherical analyzer gives spectral function A(EB,k||). Ekin is measured in hemispherical analyzer applying static electric field. I hope I am on the right track to learn exciting ARPES.
Thanks for your teachings.

 

[sweet springs]

Hi,

In general, I know that one measures the momentum of a charged particle by measuring its angle of deflection while passing through a magnetic field. But how does one determine such an angle without knowing the particle’s position at the same time?

Momentum measurement belongs to the measurement of the second kind. After measurements, states do not stay in eigenstates |p> any more where p are the observed values. Any precise measurement of moment is possible, but the state after measurement must suffer not zero disturbance from the eigenstate. See Wiki Measurement in quantum mechanics for the measurement of the second kind.
Regards.

 

[ikos9lives]

It isn't that you can know the position and not the momentum or vice versa. It's that both have degrees of uncertainty that are inversely proportional. i.e. the more precise the particle's position, the less precise its momentum. I don't think you would ever experimentally have a situation where you have an exact value for one and infinite vagueness for the other, though I guess that would be the extreme case.

One way you could increase the precision of momentum measurement while decreasing precision of position would be by "looking" at it with longer wavelength photons, which would disturb the momentum less but have a lower "resolution" than shorter wavelengths.

 

[sweet springs]

Hi.

It isn't that you can know the position and not the momentum or vice versa. It's that both have degrees of uncertainty that are inversely proportional. i.e. the more precise the particle's position, the less precise its momentum. I don't think you would ever experimentally have a situation where you have an exact value for one and infinite vagueness for the other, though I guess that would be the extreme case.

Let us consider two successive position measurement for a free particle of mass m, i.e.,
At T=0 rough measurement of position resulted from x1-Δx to x1+Δx
At T=t rough measurement of position resulted from x2-Δx to x2+Δx
Velocity is within [ (x2 - x1) / t - 2Δx / t, (x2 - x1) / t + 2Δx / t ]

In this experiment you can take Δx as large as you like so that momentum disturbance is as small as you like. Then we can make x2-x1 >> Δx and Δx / t << 1 by taking large t except the very rare case that velocity is precisely zero. Velocity or momentum divided by m can be measured as precise as you like.
Is anything wrong with it ?
Regards.