Explaining electron spin and Pauli exclusion principle to children

Fundamental particles are the building blocks of nature, of which the photon and the electron have the most visible impact on our everyday life. Photons are all pervasive. If they have the right energy, they can stimulate your eyes' photoreceptor cells. At other energies they will warm you up because they radiate from a warm object. Electrons are more energetic than the photons. They can either be free in space, or bound in atoms. Through their motion, they transmit motion to photons, which in turn can excite other electrons at distant places. This phenomenon, known as electromagnetism, is used in all wireless transmissions. Photons are the electromagnetic force carriers and electrons are the electromagnetic force sources.

In order to understand the behavior of photons and electrons, it is important to have analogies that help us keeping track of them. In previous posts, I mentioned some helpful analogies for photons (for example at this post on polarization). Although electrons also show wave behavior, they act a bit differently from photons. You can not stack electrons near to one another, except if they have compatible spinning motions. For spinning motions to be compatible means that the electrons must:

• either spin at rates whose proportions are expressed with integers: for example one electron spins twice as fast as the other electron,
• or spin in different directions, if they spin with the same velocity.

I sometimes come across situations that remind me of electrons. If you're standing in the bus or in the metro, you grip a pole to keep equilibrium. In the metro-train in my Paris suburb, the poles occur in pairs, like in the picture aside. When my children were younger, one of their favorite games was to spin around those poles. For parents, if you let your kids spin around the poles disorderedly, this game can be quite stressful, ending with fighting or crying. I used to explain to them that they had to spin like electrons in atoms. If one kid spins in one direction, the other kid needs to spin in opposite direction, in order to avoid hard clashes. I recently asked them if they could do it again so that I could put it on movie and post it to illustrate this electron analogy. But they've grown up and are now ashamed to play such games:-) So I decided to create the following simple animations that illustrate the electron spin and the Pauli exclusion principle.

A kid spinning around the pole is alike an electron spinning around a proton in its state of minimum energy, see Figure 1. Physicists designate the spinning direction with the help of the right hand rule. The kid of Figure 1 therefore has its spin down.

If your second kid spins in the same direction around the other pole, you can be sure that this game won't last for long. Their motions are incompatible and it ends up with a clash, see Figure 2.

If you want them to play peacefully, you need to instruct them to follow a natural rule: the Pauli exclusion principle, illustrated in Figure 3. Electrons with same spinning velocity and sharing the same space can only occur if their spins are opposite. Very useful rule to keep harmony in the family!

Follow-up of my FQXi essay: Ordinary analogues for Quantum Mechanics

Today I had the good surprise to discover the article "Quantum mechanics writ large" written by John W. M. Bush, Professor of Applied Mathematics at MIT, promoting the work of Couder, Fort et al. on the bouncing droplets. John Bush writes: "At the time that pilot wave theory was developed and then overtaken by the Copenhagen interpretation as the standard view of quantum mechanics, there was no macroscopic pilot wave analog to draw upon. Now there is."

I'm totally in line with this opinion. Quantum mechanics has macroscopic analogues which have so far nearly never been discussed and from which we would learn a lot. There has already been some discussion along with my 2009 FQXi essay. In the abstract, I wrote something similar to John Bush: "Classical physics was not sufficiently advanced to deal with macroscopic particle-wave systems at the birth of quantum mechanics. Physicists therefore lacked references to compare quantum with analogous macroscopic behaviour. After consideration of some recent experiments with droplets steered by waves, we examine possibilities to give some intuitive meaning to the rules governing the quantum world."

So, I hope this new article will gain much attention and foster discussion about macroscopic analogues for quantum behavior.

Feynman and what comes next...

As you may be aware of, I am a Feynman aficionado:

1. My scientific motto is a (not so famous) quote of Feynman: "All we do is draw little arrows on a piece of paper - that's all!"
2. My Twitter visual profile is dedicated to him.
3. I love viewing videos of his lectures or his interviews.
4. I regularly go across his written works.
5. I've spotted errata in his Physics Lectures, volume III (Quantum mechanics), most of them typo, but some substantial errors.

Curiously, I discovered him relatively late. When I followed quantum mechanics courses at university (somewhere between 1985 and 1989), my courses didn't refer to his lectures. I consider that as a missing. It was only after I took time to dig deeper into the quantum foundations (after 1996) that I came across his works. Reading his works was so enlightening for my comprehension of the fundamental laws of nature, that there are pieces that I could read tens of times and each time I would learn new things. Or better said: approach known things from a new point of view.

Feynman is deservedly one of the most quoted people (at the Selected Pages section of Wikiquote, he figures along with people like Aristotle, Buddha, Confucius, Einstein, Jesus or Shakespeare...). His words are inspiring and often explain physical truths in plain language, comprehensible to the layman. As for all quotes, there is a caveat: they must not be seen as an absolute truth. Or as Feynman stated it himself: Learn from science that you must doubt the experts.

Very early I was skeptic about one of Feynman's most famous quotes: nobody understands quantum mechanics. This is often requoted in a more or less transformed way (for example Dawkins' version: "If you think you understand quantum mechanics, you don't understand quantum mechanics"). Does this quote have a general and definitive value of truth? Or was it just that Feynman didn't know of anyone who could explain quantum mechanics in an understandable common sense way?

Chapter 1 of Feynman's quantum lectures gives some insight in the reasons of his belief that nobody understands quantum mechanics: "We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics." He goes on to describe the double-slit experiment (with electrons), showing that it is impossible to think of waves alone or of bullets alone (such explanations have been taken over by popular media like that given by "Granddaddy of all Quantum Weirdness"). And Feynman concludes with "No one has found any machinery behind the law. No one can explain any more than we have just explained. No one will give you any deeper representation of the situation." These are terrible sentences when repeated to hundreds of thousands, maybe millions of physics students since 1965. They mark a halt for any further investigation of the subject.

Fortunately, there are inventive unconventional physicists. There were already deeper theoretical representations given by physicists like De Broglie or David Bohm, showing how a particle may be directed by a guiding wave and thus yield all the experimental results of the double slit experiment, but this had never been put to proof during their lifetime with an experimental model.

Today, I think I can safely say that the quote "nobody understands quantum mechanics" is experimentally outdated. Couder and Fort, two French physicists, experimented with bouncing droplets on a liquid subtract and discovered that they exhibited quantum behaviour, without looking for any quantum analogy:

• droplet travelling in its wave,
• diffraction and interference patterns of travelling droplets similar to photon and electron diffraction patterns,
• attraction and repulsion of droplets embedded in their waves,
• symmetric and anti-symmetric orbital motion of droplets.

Visuals presented by Couder are breathtaking. Even if you don't understand french, I highly recommend watching bouncing droplets orbiting around each other (for example at 25:35 of his 2006 presentation).

An upcoming paper of Couder's group in Physical Review Letters even suggests a quantum tunneling analogy with ordinary droplets: "Unpredictable tunneling of a classical wave-particle association", by A. Eddi, E. Fort, F. Moisy, and Y. Couder.

So today, Feynman's defeatist words about nobody understanding quantum mechanics are outdated. Please, experimental physicists, go ahead, be inventive and focus on experiments where ordinary macroscopic individual particles simulate quantum behaviour, polarization, bosonic and fermionic behaviour, inward bound forces, entanglement, quantum erasure, coupling of ordinary particles to their pilot-wave fields (gravitation, electromagnetism). Because all these quantum phenomena may be rationally understood with the help of experimental models. It's just a matter of inventivity. And we "will find someday that, after all, it isn't as horrible as it looks." ~ Feynman's Epilogue to his Lectures on Physics.

Quantum probabilities with ordinary objects

It is commonly asserted that quantum probability distributions may not be obtained with ordinary everyday objects. For example, in his sixth Stanford lecture on Quantum Mechanics, Professor Leonard Susskind says that "it is quite hard to think of a classical setup that would produce the same kind of probability distributions and how they depend on the orientation of the polarizers. I don't think anybody has ever successfully designed such a thing, at least not in any great generality that would do the right thing to polarized photons". There have already been some tries to design such experiments, for example with Aerts' Quantum Machine. I have always wondered why there has not been more research on such models, especially using Bohmian pilot-waves. The following video gives a proposal for such a setup with ordinary objects.



Here is the videoscript:
Hello, I’m Arjen the Common Sense Quantum Physicist. This sequence is a video comment on a point mentioned by professor Leonard Susskind in his sixth lecture on QM. I highly recommend this lecture for those who want to be introduced to QM and to the way how quantum state vectors are used to compute measurement probabilities on the polarization of photons. At one point in this lecture, at about the 12th minute, professor Susskind asserts that “and it is quite hard to think of a classical setup that would produce the same kind of probability distributions and how they depend on the orientation of the polarizers. I don't think anybody has ever successfully designed such a thing, at least not in any great generality that would do the right thing to polarized photons”. Here it is commonly accepted that no classical model could reproduce the full outcome of polarization measurements on photons. There are no means to obtain two distinct outcomes with the quantum probabilities cos² theta and sin² theta for instance with flying bullets that are flagged with a direction theta. However, and here is the point I want to put forward, if one makes use of so called pilot-waves, it is possible to reproduce the quantum probabilities with a model that uses ordinary everyday objects. Pilot-waves were introduced by Louis de Broglie in the 1920's in order to make some intuitive sense of Quantum Mechanics and were later rediscovered by David Bohm. Pilot-waves have the ability to steer the motion of particles. So let’s see how we could design such an experiment with ordinary objects and that reproduces the quantum probabilities of polarized photons.

We start with an object that we may describe with the help of a quantum state vector, for instance a spinning needle of unit length that we shoot in the z-direction. For reasons of simplicity, we restrict the example to the situation where the needle spins in a plane parallel to the z-direction. If the needle spins parallely to the x-z-plane, we denote its state vector by |x> or (1 0). If the needle spins parallely to the y-z-plane, we denote its state vector by |y> or (0 1) and if the needle spins in an arbitrary plane of angle theta with the x-z-plane, still parallel to the z-direction, we denote its state vector |theta> by cos(th) |x> + sin(th) |y>. So if th=0, we check that |theta=0> = 1.|x> + 0.|y> = |x>. The same for theta=90°, we check that |theta=90°> = 0.|x> + 1.|y> = |y>. So we’ve assigned a quantum state vector to the spinning state of an ordinary object. So that's the easy part!

The next step is to describe a two-valued measurement R_M where one value is obtained with probability cos²th and the alternative value with probability sin²th if the needle is in state |theta>. Let us use a wire-grid whose wires are spaced by the length of the needle and put it on the path of the needles, perpendicularly to the propagation direction. Let us then define the result RM = +1, if the needle passes through the grid without touching any wire: then it wins a point. If the needle touches a wire it looses one point: the result of the measurement will be R_M = -1.

We are considering the ideal case where the needle and the wires of the grid are infinitely thin. We may do this because this is a thought experiment. In real life, needles and wires always have a finite thickness and that would change a bit the results of the experiment. But for the sake of simplicity let us ignore that. Let us now fix the direction of the wire grid such that the wires are vertically aligned along the x-direction.

So, if the needle is in the |x> state, that is if it is spinning parallely to the x-z-plane, there is zero probability for that needle to touch a wire of the grid because we are considering the ideal case of infinitely thin needles and wires. The result for a needle with th = 0 is always +1. So there is a probability cos² th = cos²0 = 1 that it will pass the grid unaffected. So we have a probability 1 for that possibility.

If the needle is in the |y> state, that is if it is spinning parallely to the theta = 90° y-z-plane, we want the needle to have cos² theta = cos²90° = zero probability to pass the grid unaffected. So we want it to always touch a wire of the grid. Well, remembering that the spacing between the wires is equal to the unit-length of the needle, this may be achieved if the needle always has an angle of 90° with the z-axis when it arrives at the plane of the wiregrid. For theta = 90°, we need a kind of pilot-wave that steers the spinning motion of the needle in such a way that the phase of the needle with respect to the z-direction is always 90° when it arrives at the plane of the wire-grid.

And if the needle is in the |theta> state = cos(th) |x> + sin(th) |y>, we want the probability to pass the grid unaffected to be cos²(th). Equivalently this means that the probability to touch a wire of the grid must be 1-cos²(th) = sin²(th), which physically means that the length of the projection of the needle on the y-axis must be sin²(th) when it arrives at the plane of the wire-grid. Well, we can verify that this is arranged if the pilot-wave steers the needle in such a way that the phase of the needle with respect to the z-direction is always theta (or 180°-theta) when it arrives at the plane of the wire-grid.

So we've managed to retrieve quantum probabilities with ordinary spinning needles, provided that the orientation of the needles is steered by a pilot-wave. Building such an experiment is not trivial, but with some creativity it is in principle possible. At least someone could simulate it with some nice computer animation. This would allow easier visualization of quantum processes.