# Experimentally measure velocity/momentum of a particle in quantum mechanics

In the context of quantum mechanics one cannot measure the velocity of a particle by measuring its position at two quick instants of time and dividing by the time interval. That is, $$v = \frac{x_2 - x_1}{t_2 - t_1}$$ does not hold as just after the first measurement the wavefunction of the particle "collapses".

So, experimentally how exactly do we measure the veolcity (or say momentum) of a particle?

One way that occurs to me is to measure the particle's de Broglie wavelength $\lambda$ and use $$p = \frac{h}{\lambda}$$ and $$v = \frac{p}{m}$$ to determine the particle's velocity. Is this the way it is done? Is there any other way?

My bachelor thesis was about particle identification at ALICE experiment, so I can try to give you some basics.

Your intuition is right. You can find the velocity of a particle using $v=p/m$ but you can see that we just shifted the problem: how do we measure momentum? But most important: how do we measure mass? How do we identify the particle?

In ALICE Experiment the detectors are surrounded by a magnet which produces an homogeneous magnetic field (up to 0.5 T). As you probably know, charged particles in a magnetic field are deflected and this leads to the measure of their momentum, since $p=qRB$, where $q$ is the electric charge, $B$ is the magnetic field and $R$ is the curvature radius.

As you can see, we just shifted the problem again: we have to determine the curvature radius. This can be achieved thanks to $\textit{tracking}$ detectors, whose main task is the reconstruction of the particle $\textit{track}$ or path.

For example, the main $\textit{tracking}$ detector of ALICE is the TPC (Time Projection Chamber), a cylinder-shape detector full of gas. The charged particle, passing through the gas, ionizes its atoms and the result will be a "track" of electrons drifting towards the readout channels thanks to the homogeneous electric field of the TPC. This oversimplifies things a bit but at least you get an idea. Truth is the full $\textit{tracking}$ of the particle is achieved combining data of detectors, using fitting methods (such as Kalman Filter), etc.

But we have one last problem. We may know everything about the momentum and track of the particle. But how do we identify our particle, determine its mass and finally find its velocity? We can't say if our particle is a kaon, a pion or a proton just by knowing its momentum.

Fortunately, we know that a particle of momentum $p$ and mass $m$ takes a specific time $t$ to cover a distance of length $L$: $$t=\frac{L}{c}\sqrt{\frac{m^2c^2}{p^2}+1}$$

The measurement of the time of flight of the particle in ALICE is achieved by the TOF detector (the detecting element is a so-called MRPC, which picks up signals caused by electron showers coming from the ionization of the MRPC gas). Knowing $t$, $p$ and $L$ we can get to the mass $m$, identifying our particle. And finding thus our velocity.

(Of course to determine $t$ you have to know the start-time $t_0$, at the vertex point. But that's another -long- story)

• Thank you so much for this—this is just the beginnings of the information I was looking for. Can you suggest any further reading on experimental techniques for particle measurement like this, past and present? – T3db0t Apr 18 '18 at 17:20
• This doesn't answer the OP's question. ALICE (and any tracking detector) is making measurements of position in order to determine the radius of curvature and thus momentum, and does not practically get anywhere close to running up against the HUP. The position measurements of the tracker cause the momentum to spread out, thus making this method problematic given OP's concern. In the case of ALICE, the position measurements are far too crude to cause the momentum to spread out appreciably, but in principle the question still stands: how can p be measured without measuring the non-commuting x. – user1247 Feb 20 at 17:30

A particle's velocity can be measured just as you've described. If you shoot the particles through apertures that are much larger than their wavelength, the wavelike effects are minimal and they continue with basically the same momentum. If you squeeze the target aperture, though, particles that pass through continue with the same speed, but in different directions. This is what the position/momentum uncertainty means - as the aperture narrows, we know more and more about where that particle is when it goes through. Therefore, we will know less and less about which way it goes afterwards.

There are many ways to measure the energy of a particle. You're right to recognize the relationship between the wavelength and the momentum, but these values are related algebraically. The energy and momentum are related (for massive and mass-less particles) by $E^2 = (p c)^2 + (m c^2)^2$. While we can measure the speed of photons in a vacuum, that is a defined unit, so we're really measuring the length of a meter when we perform that experiment.

• I got your first paragraph. But the second paragraph was a bit confusing. Firstly, how does one measure frequency of a massive particle then? And secondly, when you say "while we can measure the speed of photons in a vacuum" do you mean to say we can measure the wavelength and frequency of photons and thus infer the speed or is there some special method to measure the speed of a photon? – Abhijeet Melkani Feb 26 '17 at 5:46
• Energy is the easiest thing to measure, and I should have said that. Photons travel at the speed of light in a vacuum. Because we define the speed of light, we really measure the length they travel. – user121330 Feb 27 '17 at 18:37
• Thanks for the response. It was very useful. But as the bounty can be awarded to only one person I think Luthien's answer is more deserving. – Abhijeet Melkani Mar 3 '17 at 5:04

For a cold atom experiment, experimentalists use time-of-flight (TOF) measurement to determine the momentum distribution of atoms in the optical trap. Suppose there are an ensemble of atoms trapped in the optical trap, when the optical trap is switched off, the atoms will "fly around" with their momentum. With detectors installed around the trap, one could obtain both the value and direction of atomic momenta, which could be gathered to contruct the momentum distribution.

• So the detectors measure the kinetic energy (and thus magnitude of momentum) and the angle between the detector and the trap gives the direction of momentum? – Abhijeet Melkani Feb 26 '17 at 5:48
• Well, the principle is rather simple, one only need to know when detectors would receive a signal (since there are usually thousands of atoms in the optical trap, there might be many signal peaks that hit the detectors) (TOF determines the "velocity"), and which detector atoms hit (which determines the angle of the momentum). Then the momentum distribution can be constructed, and after Fourier transformation, the spatial distribution can be obtained too. – Exhaustive Feb 26 '17 at 9:25
• "I'm not aware of any experimental method to detect the de Broglie wavelength of particles." Every diffractive scattering experiment tells you the quantum wavelengh of the subject and for massive particles that is the de Broglie wavelength. But, of course, diffraction negates the understanding of position. – dmckee Feb 27 '17 at 20:46
• Thanks for the response. It was very useful. But as the bounty can be awarded to only one person I think Luthien's answer is more deserving. – Abhijeet Melkani Mar 3 '17 at 5:04

Your method for measuring observables is perfectly good but there are many other ways to measure observable quantities.

Firstly, there is no perfect way to measure these observables, but the most commonly used one is to measure its deflection when it is passing through a magnetic field. In cloud chambers, charged particles are passed through a magnetic field of known strength $B$. Using the formula $R=\frac{p}{qB}$, where R is the radius of the circle that is formed when the charged particle moves into a magnetic field, the momentum and velocity can be calculated. This method is used in many places like CERN.

Even though this method works only for charged particles, most particles in the Standard model are charged and deflect when they are passed through a magnetic field.

EDIT 1: For specific observables however, there are certain experiments such as for Spin there is the Stern-Gerlach experiment.

Hope this helps

• Thanks for the response. It was very useful. But as the bounty can be awarded to only one person I think Luthien's answer is more deserving. – Abhijeet Melkani Mar 3 '17 at 5:05

In condensed matter physics community, one can use the ARPES apparatus. ARPES gives information on the direction, speed and scattering process of valence electrons in the sample being studied (usually a solid). This means that information can be gained on both the energy and momentum of an electron, resulting in detailed information on band dispersion and Fermi surface.