When can a single particle be treated classically?

I feel like after 15 years of doing physics I ought to know the answer to this question, but I don't. Here are two variants that will help describe my confusion:

1) When can a single, free, non-relativistic particle be treated classically?

I assume the answer is "never", since wave packets spread so fast. So, while the expectation value of the position of the particle moves in a straight line like a classical particle, the probability of finding it on the classical trajectory is basically zero.

2) When can a single charged particle in an external electromagnetic field be treated classically?

I would have also guessed "never" by similar reasoning to point 1. However, textbooks are full of statements about electrons following the Lorentz force law. Wikipedia even has a picture of electrons in a "teltron tube". Every plasma physics textbook stats with N classical particles (electrons/ions) interacting by the Lorentz force. What is going on? What is the justification for this approximation? Is it secretly some statistical statement valid only for large N (i.e. the words are wrong but the kinetic equations are correct)? Is there any reference that discusses this? I cannot see how a single particle can ever be treated classically.

Note: I am not asking why macroscopic bodies behave classically. That's an interesting question but there is a lot of work on it. I am also not asking about special configurations like coherent states in a harmonic potential. I am asking about single (say, elementary) particles in generic situations.

  • $\begingroup$ As you indicated, strictly speaking, "never", unless a huge number of quanta were involved, and even then there is formal trouble. That's how QM was discovered: by studying particles. This is a dangerous frontier/interface, where paradoxes run to hide... Fast particles may be localized and produce bubble chamber tracks, and cathode ray electrons may be described by dispersive wave packets, to be sure, but.... caveat lector. $\endgroup$ Dec 22 '19 at 15:49
  • $\begingroup$ ... so, loosely speaking, "sometimes", if the respective energies are sufficiently high. I would plug in the numbers in a wave packet. High-energy particle beams are almost classical, with substantial quantum corrections (a quantum halo). Ultra high-energy particles in HEP and cosmic rays are as classical as BB pellets: The $\Delta x$s imposed by the UP are fantastically small, so the can be visualized as concentrated on a classical trajectory. $\endgroup$ Dec 22 '19 at 16:08
  • $\begingroup$ Near duplicate. $\endgroup$ Dec 22 '19 at 16:11
  • $\begingroup$ It's not a duplicate. The other post asks about a beam, whereas I ask about a single particle. What's a "BB pellet"? Also, perhaps we can restrict to a non-relativistic particle just to sharpen the question. Is all of non-relativistic plasma physics wrong? (Obviously not, but why?) $\endgroup$
    – Sam Gralla
    Dec 22 '19 at 17:25

I'm clearly out of my (inverse) depth on this, as a plasma physicist or a beam physicist would be much more qualified to provide you with their fields' conventional wisdom on this.

But I'd be a traitor to my academic great-grandfather if I didn't even try to reassure you and myself (!) that a tiny "marshmallow" (not BB after all) electron in a CRT (old TV set like mine) does not spread in its freely-propagating wave packet into something bigger that would compromise its effect on the mask & TV screen! So, this is a narrow back-of-the-envelope reassurance cartoon, to possibly orient you in the erudite discussions on broad semiclassical limits sure to emerge here.

The other electrons in the beam do not affect it much. It boils off the cathode at a temperature, let us say ~1160K, so, then, 0.1 eV. It picks up a kinetic energy of about 0.5keV, so 3 orders of magnitude smaller than its mass of 500keV. Its speed is then less than 5% of c, quite non-relativistic.

Now, modeling the marshmallow with a Gaussian wavepacket (best described in Pauli's sublime and concise Wave Mechanics lectures' booklet), we wish to check that the marshmallow has not spread appreciably by the time it hits the screen, 50cm away, as dispersive wave packets are wont to do.

In specific terms, we wish to check that the final width over the initial width $\Delta x_0$ is not very different than one, $$ \sqrt{1+\left (\frac{\hbar t}{m(\Delta x_0) ^2}\right )^2 }, $$ that is, we seek reassurance that the large parenthesis of the second term is not substantially larger than one, i.e. it has not gotten to dominate, yet, as feared.

Since the velocity is 0.05c, we use HEP natural units $\hbar=1, c=1$ and eVs, and a distance to the screen of d ~ 0.5m ~ 2.5 $10^6$/eV. Now the basic assumption is that the width of the marshmallow is related to the temperature of its thermionic emission, i.e. it is 1/0.1eV. Consequently, $$ \frac{ d}{0.05~m(\Delta x_0) ^2} \approx 1 . $$

This means that the modeling wavepacket has not dispersed by the time it hits the screen: a quasi-classical behavior. (I'd thus bet CRT engineers do not agonize about quantum effects here.)

  • $\begingroup$ Thanks for writing this out, but as you say this is just about a specific situation where the electron is "measured" before it has a chance to spread. But there are many situations where classical propagation is invoked and there is plenty of spread. For example, a quick search says the solar wind has temperature 10^5K and mean free path 10^8 km. So, a "classical electron" with v~sqrt(2kT/m)~1700 km/s takes about about 16 hours to find another particle. And the "collision" is presumably electromagnetic anyway. $\endgroup$
    – Sam Gralla
    Dec 23 '19 at 15:36
  • $\begingroup$ Sure. I just reminded myself (you?) that some type of classical description may work. Cosmic and HEP physicists deal with that mental picture routinely. But, of course, in each context, the devil is in the HUP details involved. I'd assume, without much empirical support, that each field has its own seat of the pants understanding... $\endgroup$ Dec 23 '19 at 15:42

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