Physics / Short Notes

Vol. 3, NO. 3 / November 2017

Three for the Road

Jeremy Bernstein

Letters to the Editors

In response to “Three for the Road

Max Born published a brief paper entitled “Zur Quantenmechanik der Stoßvorgänge” (On the Quantum Mechanics of Collisions)” in 1926.1 This was one of the seminal papers in the history of the quantum theory. That same year, Erwin Schrödinger published the first of his papers on wave mechanics. It was not clear how to understand the wave function. A natural interpretation, at first endorsed by Schrödinger, was to think of waves as Führungfelder (guide fields) for particles, but this interpretation rapidly fell out of favor when it seemed to predict that the electron in the hydrogen atom could be guided to the moon. Born had studied the collisions between an electron and an atom. He suggested interpreting the wave function as the probability that the electron was scattered in some particular direction. But then Born corrected himself. It is the square of the wave function that is related to the probability.

This was an interpretation that certainly did not seem obvious on its face. At the 1927 Solvay Conference, Albert Einstein raised an issue, one recounted by Born:

A radioactive sample emits α-particles in all directions; these are made visible by the method of the Wilson cloud chamber. Now, if one associates a spherical wave with each emission process, how can one understand that the track of each α-particle appears as a (very nearly) straight line? In other words: how can the corpuscular character of the phenomenon be reconciled here with the representation by waves?2

Born responded in terms a modern quantum theorist would find familiar:

As soon as such ionization is shown by the appearance of cloud droplets, in order to describe what happens afterwards one must reduce the wave packet in the immediate vicinity of the drops. One thus obtains a wave packet in the form of a ray, which corresponds to the corpuscular character of the phenomenon.3

This left open two questions. Could this collapse be described within quantum theory itself? And why should the path be more or less a straight line?

In 1929, Nevill Mott published a paper entitled, “The Wave Mechanics of α-Ray Tracks.”4 Take the inelastic scattering of an α-particle with a hydrogen atom. The particle’s wavelength is 10–15 meters; the hydrogen atom is about 10–10 meters across. The final α momentum is k. We neglect the excitation energy of the hydrogen atom, so the scattering is elastic. The amplitude in the Born approximation is proportional to ψfinal*Vψinitial, where V is the Coulomb potential. The initial wave function is proportional to eikR/R, where R is the position of the α-particle. The final wave function is proportional to an outgoing plane wave. But if the vector that describes the location of the hydrogen atom in question is called a then the Coulomb potential is only large when R~a. If we make the various substitutions in the transition amplitude, we find an integral which has a factor eikR(1–ka). Here k and a are unit vectors, and ka is their scalar, or dot, product. If ka = 1, all is well; if not, extreme oscillations appear over the electron orbital, effectively making the scattering amplitude zero. The final α must be in the direction of the initial one. We are in the process of generating a straight line. Mott shows this for two successive scatterings; no doubt one could continue this indefinitely.

But does this answer the question? Not exactly.

It surely will have struck the reader that there is no wave function collapse here. This is not surprising. Schrödinger’s equation can never produce a collapsed wave function. Time evolution is a unitary operator and projections are not unitary. The ionization of some hydrogen atoms by collision with α does not produce an observable track. Bubbles are needed, condensates around the ions produced by the α-particle collisions. This process, which accounts for the cloud chamber tracks, is in practice irreversible; it very likely cannot be described using ordinary quantum mechanics.5

The Mott analysis was necessary, but not sufficient.6

Schrödinger had become skeptical of the foundations of quantum mechanics by 1935. If Paul Dirac had had any such qualms, he wisely kept them to himself. Schrödinger shared his feelings with Einstein who, in this respect, was a kindred spirit.

A particle’s spin represents its intrinsic angular momentum. Aligned with a given coordinate axis, it can be either up or down: ↑ or ↓. Two electrons might be in a configuration of ↑↑, which has a total spin of one. But they also might be in a configuration of ↑↓–↓↑. In this configuration, the electrons do not have a definite spin, and we say, using Schrödinger’s language, that the spins have become entangled. So long as we do not interfere with this state by performing a measurement, the entanglement persists, no matter how far the electrons are separated in space. But this has a consequence that Einstein referred to as a spooky action at a distance.

John Bell suggested another example to me. Consider identical twins separated at birth. When, later in life, they are reunited, it is discovered that they have very similar psychological profiles. Why? Because they have the same genes. If you believe quantum theory, however, there is no analogous explanation for the persistence of entanglement.

Take the two electrons in the configuration ↑↓–↓↑, and let them fly off in opposite directions. On the moon, an observer measures the spin of the arriving electron, and on the Earth, another observer does the same thing. There is no communication between the observers; only later do they compare notes. When they do, they find a remarkable correlation. Whenever the first observer has found ↑, the second observer finds ↓, and vice versa. If they have chosen axes separated by an angle θ, then quantum mechanics predicts a correlation that is proportional to ‑cos(θ). If, during the separation, one of the observers chooses to change the angle, the correlation will reflect this. No signal has passed between the two observers.

The spin-up and spin-down probabilities are, respectively, cos(θ/2)2 and sin(θ/2)2, which, of course, sum to 1. The anticorrelation of the two spins is sin(θ/2)2 – cos(θ/2)2 = –cos(θ). Note that, when the angle is zero, the spins are perfectly anticorrelated, and when it is ninety degrees there is no correlation. No local hidden variable theory can reproduce these results for all angles. It is quantum mechanics pure et dure.

This argument can be reproduced in a simple form. I have at my disposal what I call Einstein robots. These can be programmed to do anything, except to communicate with each other at speeds greater than the speed of light. Can I program them to reproduce the quantum correlations? Imagine I have a bucket of spin ½ particles in singlet states—all electron spins are paired. I instruct the robots to take these particles and to bring them in pairs to two separate detectors. I have their magnets lined up; this setup will allow the robots to reproduce the anticorrelation of quantum mechanics. Then I rotate one of the magnets through θ. The robots have been programmed to reproduce the quantum correlation –cos(θ) when one of them encounters such a rotation. But now I play a trick on the robots. I rotate one detector by θ and the other by –θ. The robots cannot communicate, so each produces a correlation. When combined, the two give the correlation –2cos(θ).

The correct quantum correlation is –cos(2θ).

No signal can travel faster than the speed of light; one cannot claim a causal connection in the usual sense. One can note that this result is to be expected because of entanglement, but it cannot be explained in the same way as for the identical twins. The spins were entangled when the state was created and remained so until measurement. This is what Einstein and Schrödinger found impossible to swallow. Einstein spoke of spooky action at a distance, but this language is a relic of classical physics. You cannot explain quantum mechanics in terms of classical physics, although the converse is true.

Electrons are just one example of a particle with spin. All elementary particles have spin. This includes the photon, which has spin 1. As the photon propagates, its associated electric and magnetic field circulates around the direction of propagation. This is known as a state of circular polarization. The same trick done for pairs of electrons can be done for pairs of photons. The result is again an entangled state. This was shown by a remarkable experiment performed at the University of Science and Technology of China in Shanghai. At an altitude of five hundred kilometers, there is a Chinese satellite with a light-altering crystal on board. A laser signal was sent from the earth and produced pairs of polarized photons. These were transmitted to widely separated stations at high altitudes in Tibet. Their polarization remained correlated, just as quantum mechanics predicts. There is no explanation that would have satisfied Einstein. It is just quantum mechanics.

At the turn of the twentieth century, Max Planck introduced a set of natural units based on the fundamental constants of physics. Given in meters, the Planck length is

P=ħG/c3=1.616 × 10-35

where G is the gravitational constant. Planck’s work was done before the advent of special relativity, so no one asked how this length transforms. But since fundamental constants are Lorentz invariant, it doesn’t. How then can it be a length?7 I once posed this question as a riddle. Only Freeman Dyson answered that this length is not measurable.8 Suppose that there are two objects floating in space. Can they maintain their Planck distance long enough for it to be measured? Suppose the variation in the distance between them is of order δ, and suppose that a measurement of the distance takes a time T. The uncertainty in the momentum is of order δM/T, so Mδ2 ≥ ħT. Let δ equal the Planck length (Għ/c3)3/2. If the distance between the two objects is D, then T must be greater than D/c in order for the two objects to communicate. It follows that

D ≤ GM/c2.

The right-hand term should be familiar. It is the Schwarzschild radius of a black hole. The distance required to make the measurement is inside the black hole and thus unmeasurable. A similar sort of argument can be made for the Planck time. These units are not measurable. They are just numbers.


  1. Max Born, “Zur Quantenmechanik der Stoßvorgänge” (On the Quantum Mechanics of Collisions), Zeitschrift für Physik 37 no. 12 (1926): 863–67. For a very nice discussion of this early history, see Rodolfo Figari and Alessandro Teta, “Emergence of Classical Trajectories in Quantum Systems: The Cloud Chamber Problem in the Analysis of Mott (1929),” arXiv:1209.2665 (2012). 
  2. Quoted in Rodolfo Figari and Alessandro Teta, “Emergence of Classical Trajectories in Quantum Systems: The Cloud Chamber Problem in the Analysis of Mott (1929),” arXiv:1209.2665 (2012): 6. 
  3. Quoted in Rodolfo Figari and Alessandro Teta, “Emergence of Classical Trajectories in Quantum Systems: The Cloud Chamber Problem in the Analysis of Mott (1929),” arXiv:1209.2665 (2012): 6. 
  4. Nevill Mott, “The Wave Mechanics of α-Ray Tracks,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 126 (1929): 79–84. The paper is unnecessarily complex for the point being made here. Instead I will use Steve Adler’s treatment—and notation—which makes the matter more transparent. I thank Steve Adler for sending me his paper and for discussions. The paper can be found online under the title “Mott Simplified” in the talks and memos section. 
  5. For a review, see Angelo Bassi and Giancarlo Ghirardi, “Dynamical Reduction Models,” Physics Reports 379 (2003): 257. 
  6. As far as I know, there is no relativistic covariant theory of quantum mechanical measurements. This may be the real measurement problem. 
  7. The same question can be posed for the other Planck units. 
  8. Dyson discussed this in the context of the LIGO experiment but his argument does not make use of this. For details, see Freeman Dyson, “Is a Graviton Detectable?” International Journal of Modern Physics A 28, no. 25 (2013), doi: 10.1142/S0217751X1330041X. 

Jeremy Bernstein is Professor Emeritus of Physics at the Stevens Institute of Technology.

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