Perplexed by Pauli

On August 15, 2014

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.

atoms copy

The 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.

wavefunction copy

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.

 

_ _ _

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.

 

About Philip Moriarty

Philip Moriarty is a Professor of Physics at the University of Nottingham, where his research focuses on nanoscale science. He is a member of the Science Board of the Institute of Physics and coordinates the multi-partner ACRITAS European network. He has participated in a number of research council-funded public engagement projects, including Giants of the Infinitesimal, and was a member of the Programme Committee for the controversial “Circling the Square: Research, Politics, Media, and Impact” conference held in Nottingham in May 2014. He is also a regular contributor to the Sixty Symbols video series.

13 Responses to Perplexed by Pauli

  1. Before trying to understand how atoms interact, why not first understand how two electrons interact? For this kindly let us know,
    (a) Shape and size of an electron at rest.
    (b) Shape, size and nature of the electrostatic field of an electron at rest.
    (c) Shape, size and nature of wave function of an electron at rest.

    Further we also need following clarifications.
    (d) Is the electron spin similar to the rotational motion of a spinning top?
    (e) If two electrons at rest are separated by distance d, does their electrostatic interaction takes place through mutual exchange of some virtual photons or through superposition of their electrostatic fields?
    (f) Does the Coulomb force between two electrons separated by distance d depend on the relative orientations of their spins? If so, how?

    • I know this much about spin – the particle’s not spinning. It’s a property named after the effect charged particles have when they spin. If you have a charged something and spin it, it produces a magnetic moment. In the quantum mechanics, charged particles like electrons have a magnetic moment. This means that they are acting as if they were spinning, but they’re not (as far as I’m aware at any rate). If it were as a result of the electron spinning, you couldn’t separate the spin from the electron (See http://www.bbc.co.uk/news/science-environment-28543990)

    • Perhaps I can volunteer something:

      The electron’s “shape” is spherical, but it has a toroidal topology. It doesn’t have a size just as a seismic wave doesn’t have a size, but it does have a wavelength. And it doesn’t have an electrostatic field, it has an electromagnetic field. Or should I say it is electromagnetic field. Or wavefunction, which IMHO is best through of as four-potential. Remember Einstein said “a field is a state of space” and remember the frame-dragging of gravitomagnetism. To visualize the electron’s electromagnetic field imagine you start with a gin-clear ghostly elastic standing in for space, with lattice lines drawn in it. Reach into the lattice with your right hand and turn it clockwise. Then reach in sideways with your left hand and turn it clockwise again.

      Electron spin isn’t much like the rotational motion of a spinning top. It’s better to start with a tornado. It has “intrinsic” spin that makes it what it is. Without it, the tornado is just wind. Without its intrinsic spin, the electron is just light. Only it’s a “bispinor”, like a tornado of light rotating horizontally and vertically. To visualise the centre of the frame-dragged space, see Adrian Rossiter’s Antiprism website. Think Dirac’s belt, which is likened to a Moebius strip. Then mentally “inflate” it to a ring torus with two orthogonal rotations, then inflate that to a spindle-sphere torus, like this. But note that there is no actual surface, this animation is little more than a depiction of the eye of the storm, as it were.

      The Coulomb force doesn’t depend on the relative orientations of their spins, because there is no orientation for a bispinor “standing” wave with biaxial rotation. Imagine you have a glass-clock, and you can see the hands moving. You might say there’s a clockwise rotation, but walk round the back and it’s an anticlockwise rotation. Now spin the clock like a coin. Which way is it spinning? You can’t say.

      • John, it is a logical contradiction to say that electron’s shape is spherical, but it doesn’t have a size. There cannot be a spherical shape without a size. After resolving the issue of size, I would like to specifically discuss as to how exactly two electrons, separated by distance d, interact? Is there any exchange of virtual particles involved in the interaction process?
        GSS

        • Think of flat calm gedanken ocean. Now put a whirlpool in the middle. This whirlpool is round. Circular. But how big is it? You put floats in the water to detect the circular current, which is weaker as you move away from the centre. But there is no edge to it. Think of the electron as a spherical “bispinor” version of this, featuring electromagnetic stress-energy in a closed Dirac’s belt path rather than fluid flow into a sink. Check out the Falaco soliton for a vortex without a sink.

          Two electrons interact in a vorticial fashion: co-rotating vortices attract, counter-rotating vortices repel. There is no actual exchange of virtual particles, the virtual particles are “field quanta”. Think of positronium. The electron and positron attract and swirl around one another for a picosecond. Whilst they do their opposite fields largely cancel one another as far as the distant observer is concerned. It’s like they’ve swapped little bits of opposite field. But they aren’t actually throwing photons at one another. Hydrogen atoms don’t twinkle and magnets don’t shine. See Matt Strassler’s article on virtual particles and the screw nature of electromagnetism.

      • Sorry, your book chapter doesn’t answer any of the fundamental questions asked in my previous post.

        In my opinion, invoking PEP in the discussions regarding interpretation of DFM results, is neither relevant nor necessary. Basically, you don’t have to treat the electron distribution around atoms or molecules as free electron gas. These electrons surrounding atoms and molecules are all bound electrons, bound in specific orbits around the nuclei. These electron orbits, apart from displaying slight flexibility in adjusting their orientations during external interactions, act as quite rigid structures in resisting external pressures during atomic or molecular interactions as in DFM. You simply have to correlate the measured force results in DFM with the corresponding, time averaged, electron density in the test sample.
        GSS

        • No, sorry, but the entire point of the paper to which I link in the post above (and a significant amount of the book chapter to which I also link) is that one *cannot* correlate the measured force with the electron density in the test sample (as you put it). The appropriate line from the abstract to our paper (which is open access and so available for free) is:

          Interpreting DFM images of hydrogen-bonded systems therefore necessitates detailed consideration of the coupled tip-molecule system: analyses based on intermolecular charge density in the absence of the tip fail to capture the essential physical chemistry underpinning the imaging mechanism.

          You also say: “…electron orbits act as quite rigid structures in resisting external pressures during atomic or molecular interactions as in DFM.”

          Where’s your evidence for this statement? For one thing, it flies in the face of the work we reference in that book chapter (which does indeed address the fundamental questions you raise — you need to chase up the references cited in the chapter).

  2. “Where does the repulsive force due to Pauli exclusion come from?”

    From their “spinor” nature. The electron is something like a 3D cyclone, see Electrons in magnetic field reveal surprises on physicsworld. Counter-rotating vortices attract, and co-rotating vortices repel.

    People say the electron’s intrinsic spin can’t be classical because if it was rotating like a planet it would be rotating faster than light. But that’s a non-sequitur. A spin ½ particle isn’t rotating like a planet, it’s rotating like Dirac’s belt, like a bispinor. There’s a rotation at c AND an orthogonal rotation at half that rate, and the AND acts as a multiplier. Besides, the Einstein-de Haas effect says the rotation is real, as does magnetic dipole moment. Note that it was Thomson and Tait who coined the phrase spherical harmonics, and that you can read something about them in On Vortex Particles by David St John.

  3. Hi,
    Very interesting post. I just have a couple of questions:

    1. Is it the fact that the K.E. has changed that means there (appears) to be a force between the two particles? [Not sure whether appears is appropriate or not based on the comment “…of course, the force due to Pauli exclusion isn’t really a force like any other."]

    2. In what respect does the wave function changing prevent them overlapping?
    I can’t really explain the second question well. I’m guessing the left hand graph shows the two particles coming closer together pre-PEP kicking in, then the right hand side shows how they avoid overlapping, but why does that shape occur? What does the resulting graph look like is possibly what I’m asking?

    If I need to clarify the second question, please say. I’m about to start A2, and I’m aware of what the PEP is, but not exactly it’s effect outside of what’s taught in AS chemistry.

    Anyway, thanks.
    -Tom

    • You weren’t asking me were you Tom? Just in case:

      1. No.

      2. It adopts a closed path. Think of it like this: two waves can ride over one another, but two whirlpools can’t overlap.

    • Hi, Tom.

      Sorry for the delay in responding — I’m on holiday at the moment.

      1. The force arises from the change in the momentum distribution. See Section 1.5 of the book chapter and references therein (particularly the Blaylock and Mullin reference for a thought-provoking, if contentious, discussion). If you can’t access that reference please send me an e-mail and I’ll send it you (philip.moriarty@nottingham.ac.uk)

      2. The sketch of the wavefunctions in the diagram is very much just a schematic — it is certainly not the only shape that can result. The key point is that the PEP will act so that the overlap integral is as small as possible. The overlap integral is, however, a concept that you won’t encounter until first year undergraduate physics or chemistry (at the very earliest — most likely second year). This is a good set of notes on atomic and molecular orbital theory which introduces the concept of the overlap integral: https://chemistry.tcd.ie/assets/pdf/sf-chemistry/smd/molecular-orbital-theory-the-best-of-both-handout.pdf

  4. Phil: this Discovery of Electron Spin is worth reading:

    “When the day came I had to tell Uhlenbeck about the Pauli principle – of course using my own quantum numbers – then he said to me: “But don’t you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates…”

    “…The man who never cared to believe in the spin was Pauli. And then Bohr said: “On your way home you should stop off at Hamburg and explain the factor 2 to Pauli…”

    Like I was saying above, there’s a non-sequitur wherein people say the electron isn’t rotating like a planet so it isn’t rotating at all. But it isn’t rotating like a planet, the Dirac spinor is a bispinor. IMHO it’s rotating like this.