Experimentally measure velocity/momentum of a particle in quantum mechanics

Question

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?

Answer 1

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 $L$: \begin{equation} t=\frac{L}{c}\sqrt{\frac{m^2c^2}{p^2}+1} \end{equation}

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)

Answer 2

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.

Answer 3

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.

See arXiv: 1002.2311

Answer 4

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

Answer 5

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.