Perplexed by Pauli

Wolfgang_PauliOriginally published at physicsfocus.

A rather pervasive meme claiming that nothing ever really touches anything else has been circulating on the internet for a number of years. I think, although I’m not entirely certain, that it may well have its origins in an explanation by a certain Michio Kaku. This type of explanation later formed the basis of a video, You Can’t Touch Anything, from the immensely successful VSauce YouTube channel, which has now accrued nearly 3.4 million views.

I appreciate just how difficult it is to explain complicated physics for a general audience (see, for example, this article in the education issue of Physics World published earlier this year). And I also fully understand that we all goof at times – particularly, and especially, me. But Kaku has got form when it comes to over-simplifying explanations to the point of incorrectness in order to exploit the ‘Wow! Quantum! Physics!’ factor. This misleading over-simplification is similarly a hallmark of the ‘you can’t touch anything’ meme.

Why is it that this particular meme winds me up so much? (After all, there’s a universe of other, much more egregious, stuff on the internet to worry about.) Well, I think it’s mainly because it hits just a little too close to home. My research area is known as non-contact atomic force microscopy (NC-AFM) and there’s a very good reason indeed why scientists in the field draw a distinction between the non-contact and contact modes of AFM. I’ve banged on about the flaws in the meme, as I see them, to Brady Haran on a number of occasions over the last couple of years and this finally led to a video, uploaded to the Sixty Symbols channel last week, where he and I debate whether atoms touch.

If you’ll excuse the shameless self-promotion1, of all the Sixty Symbols videos I’ve done with Brady, I’m most happy with this one. It shows science as a debate with evidence, models, and analogies being thrown into the mix to support a particular viewpoint – not as something which is “done and dusted” by the experts and passed down as received wisdom to the ‘masses’. This is exactly how science should work and how it should be seen to work. (Here’s the obligatory supporting Feynman quote: “Science is the belief in the ignorance of experts”. Stay tuned – another Feynman quote will be along soon.)

The reason I’m writing this post, however, isn’t to rake over the ashes of the debate with Brady (and the associated lengthy comments thread under the video). It’s instead to address a big, and quite deliberate, gap in the video: just how does the Pauli exclusion principle affect how atoms interact/touch/bond/connect? This is an absolutely fascinating topic that not only has been the subject of a vast amount of debate and confusion over many decades, but, as we’ll see, fundamentally underpins the latest developments in sub-molecular resolution AFM.

Beyond atomic resolution

At about the same time as the Sixty Symbols video was uploaded, and entirely coincidentally, a book chapter my colleagues and I have been working on over the last couple of months appeared on the condensed matter arXiv: Pauli’s Principle in Probe Microscopy. The Pauli exclusion principle (PEP) plays an essential role in state-of-the-art scanning probe microscopy, where images like that shown on the right below are increasingly becoming the norm. Scanning probe images of this type are captured by measuring the shift in resonant frequency of a tiny tuning fork to which an atomically (or molecularly) sharp tip is attached. As the probe is moved back and forth on nanometre and sub-nanometre length-scales, the gradient of the force between the tip apex and the molecule changes and this causes a change in the resonant frequency of the tuning fork. These shifts in frequency can be converted to an image or mathematically inverted to determine the tip-sample force. Or they can be listened to.

NTCDIThe image shown above is from recent work by our group at Nottingham but I really must name-check the researchers who pioneered this type of ultrahigh resolution imaging: Leo Gross and co-workers at IBM Zurich. Leo and his colleagues first demonstrated that it is possible to acquire AFM images of molecules where the entire chemical architecture can be visualised. The images show a remarkable, and almost eerie, similarity to the textbook ball-and-stick molecular models so familiar to any scientist. Compare the experimental image of NTCDI molecules on the right above to the ball-and-stick diagram on the left where grey, blue, and red spheres represent carbon, nitrogen, and oxygen atoms respectively. These exceptionally detailed images of molecular structure2 are acquired by exploiting the repulsion of electrons due to the Pauli exclusion force at very small tip-sample separations.

I was explaining all of this to a class of first-year undergraduate students last year, stressing that the repulsion we observe at small tip-sample separations – and, indeed, the repulsion ultimately responsible for the reaction force keeping them from falling through their seats – is not simply due to electrostatic repulsion of ‘like’ charges. I wound up the lecture, chucking in the throwaway remark, “…of course, the force due to Pauli exclusion isn’t really a force like any other. You’ll cover this in quantum statistics next year.”

By the time I’d got back to my office, two email messages from students in the lecture had already made their way to my inbox: “If it isn’t a force like any other, then what the heck is it?”

That’ll teach me to be flippant with the first-years. It’s a great question. Where does the repulsive force due to Pauli exclusion come from – just why is it that electrons don’t want to be ‘squeezed’ into the same quantum state?

The Quantum Identity Crisis

Ultimately, the Pauli exclusion principle has its origins in the indistinguishability of electrons. (Well, OK, fermions – but let’s stick with the PEP in the context of force microscopy.) One frustrating aspect of the discussions of quantum statistics in various textbooks, however, is that the terms ‘identical’ and ‘indistinguishable’ are too often assumed to be synonymous. Electrons are certainly identical in the sense that their ‘internal’ properties such as mass and charge are the same, but are they really indistinguishable?

Fleicschhauer had this to say in an fascinating commentary published a few years ago:

“In the quantum world, particles of the same kind are indistinguishable: the wavefunction that describes them is a superposition of every single particle of that kind occupying every allowed state. Strictly speaking, this means that we can’t talk, for instance, about an electron on Earth without mentioning all the electrons on the Moon in the same breath.”

Well, in principle, yes, we should consider the entire multi-particle ‘universal’ wavefunction. But I’m a dyed-in-the-wool, long-of-tooth and grizzled-of-beard experimentalist. I want to see evidence of this universal coupling. And you know what? As hard as I might look, I’m never going to find any experimental evidence that an electron on the Moon has any role at all to play in a force-microscopy experiment (or a chemical reaction, or an intra-atomic transition, or…) involving electrons on Earth.

I’ll stress again that in principle, the electrons are indeed indistinguishable as there is always some finite wavefunction overlap, because there is no such thing as the infinite potential well which is the mainstay of introductory quantum physics courses. In this sense, an electron on Earth and an electron on the Moon (or on Alpha Centauri) are indeed ‘coupled’ to some degree and arguably ‘indistinguishable’. But the degree of wavefunction coupling and associated energy splitting are so incredibly tiny and utterly negligible – if I can be forgiven the understatement – that, in any practical sense, the electrons are completely distinguishable.

(Some of you might at this point be having a déjà vu moment. This is possibly connected to the (over-)heated debate that stemmed from Brian Cox’s discussion of the exclusion principle in a BBC programme a few years ago. Brian caught a lot of online flak for his explanation, some of it rather too rant-y and intemperate in tone – even for me. One of the best analyses of the furore out there is a pithy blog post by Jon Butterworth for the Guardian – very well worth a read. My colleagues at Nottingham have also discussed the controversy, and Telescoper asked if Brian had Cox-ed up the explanation of the exclusion principle.)

It is only when there is appreciable wavefunction overlap, as when the atom at the very end of the AFM tip is moved very close to a molecule underneath it (or, equivalently, in a chemical bond), that the PEP ‘kicks in’ in any appreciable way. If you want to know just why indistinguishability and the exclusion principle are so intimately connected, and how electron spin plays into all of this, I’m afraid I’m going to have to refer you to Section 1.3 of that book chapter and references therein. If you’re willing to take my word for it for now, however, read on.

Fourier and Force

Let’s cut to the chase and elucidate where that repulsive ‘Pauli force’ comes from. The PEP tells us that we can’t have two electrons with the same four quantum numbers, i.e. we can’t ‘push’ them into the same quantum state. But just how does this give rise to a force between two electrons that is beyond their ‘natural’ electrostatic repulsion? Let’s strip the problem right down to its bare bones and consider a simple gedanken experiment.

Take two electrons of the same fixed spin separated by a considerable distance from each other. We’re going to move those electrons together until their wavefunctions overlap. As the electrons get closer their wavefunctions effectively change shape so that the mutual overlap is minimal – this is the PEP in action. The figure below, adapted from a description of the origin of the exclusion principle by Julian Su, schematically illustrates this effect.


On the left hand side the electron wavefunctions are not constrained by the exclusion principle, while on the right the PEP has been ‘switched on’. The essential point is this: the exclusion principle causes wavefunction curvature to increase. Because the kinetic energy (KE) of an electron is directly proportional to wavefunction curvature via the KE operator in quantum mechanics, increased curvature means increased kinetic energy. Or – and this is the description I much prefer because it’s yet another example of the beauty and elegance of Fourier transforms – higher wavefunction curvature requires the introduction of higher spatial frequency (i.e. higher momentum) contributions in Fourier space. It this change in the momentum distribution which gives rise to the Pauli repulsion force.

While this captures some of the essence of the exclusion principle (and certainly is enough to provide important insights into what’s going on in force microscopy experiments), it doesn’t even begin to scratch the surface of the underlying physics. I suspect that Pauli himself would dismiss all of the above with his trademark “… es ist nicht einmal falsch”. He himself said in his Nobel prize lecture of 1946 that “I was unable to give a logical reason for the Exclusion Principle or to deduce it from more general assumptions…” Almost two decades later, Feynman had the following to say:

“It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is deep down in relativistic quantum mechanics. This probably means that we do not have a complete understanding of the fundamental principle involved.”

Like Feynman, I remain somewhat perplexed by Pauli’s principle.


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1 In these social-media-enabled times, I guess that shameless self-promotion has become the academic’s stock-in-trade. Perhaps, however, it was ever thus.

2 There has, however, been a great deal of controversy of late as to the origin of the intermolecular features observed in AFM images by a number of groups, including ourselves. See our chapter on the role of the exclusion principle in probe microscopy for more detail.