A Physics Challenge: Explain Pauli’s Exclusion Principle

I was recently teaching a grade 12 physics class and had to explain why only two electrons can be in the same orbital. My explanation went a bit like this (follow the links for explanations of the concepts):

Diagram showing the possible spin angular mome...

Diagram showing the possible spin angular momentum values for 1/2 spin particles (for example, electrons) (Photo credit: Wikipedia)

Electrons are spin-1/2 particles. Particles with half-integer spin are called Fermions. Fermions have a really strange property: no two fermions can be in the same quantum state. Because of this, we can only have two electrons in the same energy level: one with its angular momentum pointing down (you can imagine it spinning clock-wise on itself) and one with its angular momentum pointing up (or spinning counter-clock-wise). After that, you’re out of options: the next electron has to be in a different energy level, since electrons are only allowed two possible directions of spin


The question that followed left me at a loss for words:

Why can’t two fermions occupy the same quantum state?

The physicists amongst you will have an answer ready: because of Pauli’s exclusion principle. For those of you who never heard about it, Pauli’s exclusion principle says that no two fermions can be in the same quantum state.

However, as an explanation this is not great. Basically what we are saying is: “no two fermions can be in the same quantum state because there is a principle that says that no two fermions can be in the same quantum state.”

Again, the physicists amongst you may have another answer ready: Pauli’s exclusion principle is, after all, a consequence of the spin-statistics theorem. What happens is that particles with a half-integer spin must have an antisymmetric wavefunction. This means that, if we exchange any two particles, the sign of the wavefunction must invert.

English: Asymmetric wavefunction for a (fermio...

English: Asymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential. (Photo credit: Wikipedia)

For those of you who are not physicists, this is harder to explain. The idea is that the probability of finding a certain particle in a certain state is given by this mathematical monster called the wavefunction. We calculate probabilities by taking its square (not exactly, but close enough.) A wavefunction can describe not only one particle, but several. In fact, in quantum field theory we don’t talk about particles but fields and the particle number can oscillate. When our wavefunction is antisymmetrical, it means that, by exchanging any two particles, we get a minus sign. That is:

Wavefunction (electron 1 in state 1, electron 2 in state 2) = -Wavefunction (electron 1 in state 2, electron 2 in state 1)

Now, what happens if two electrons are in the same state? Then we have:

Wavefunction (electron 1 in state 1, electron 2 in state 1) = – Wavefunction (electron 1 in state 1, electron 2 in state 1)

However, electrons are indistinguishable particles, so electron 1 = electron 2. Therefore, we have:

Wavefunction (electron 1 in state 1, electron 1 in state 1) = – Wavefunction (electron 1 in state 1, electron 1 in state 1)

There is only one number that’s equal to its inverse: zero! This means that the wave function for two fermions in the same state has to be zero. Since the wave function is the square root of the probability, we have that the probability of finding two fermions in the same state is zero. Therefore, no two fermions can be in the same state.

However, we still haven’t explained anything. Because the next question is:

OK, but why does the wavefunction of a fermion have to be antisymmectric?

And here I have to say I’m stumped. Yes, I know how to derive this mathematically from Dirac’s equation, but I have no idea how to explain it in any mildly intuitive way. I have also been looking online for an easy-to-understand explanation and found absolutely nothing.

Hence, I want to propose a challenge for my physicist readers: can you come up with an intuitive, math-free way of explaining Pauli’s exclusion principle?

I can’t offer a prize, but I can offer my undying gratitude and a link to you in my next (and this) article.

Also, you will have contributed to enlightening a very curious 18-year-old.

That’s gotta be worth something.

Here’s a new take on the issue from my new blog.

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19 thoughts on “A Physics Challenge: Explain Pauli’s Exclusion Principle

  1. elkement

    Thanks for the challenge – but I think I’ll fail epically to explain it any better 🙂 If I recall correctly we are in good company as Feynman did, too. I can’t remember where I read that but it was in conjunction with his saying of (paraphrasing) “We don’t really understand what cannot be explained to freshmen students.”.
    So Feynman tried to come up with exactly what you ask for or something equivalent – I think the question was why electron spin is assigned 1/2. Anyway, Feynman was not able to bring the Dirac equation and fermion statistics to the masses.

    Another benchmark of mine is Sean Carroll who really went boldly where no science popularizer went before in trying to explain connection fields in his Particle at the End of the Universe. He has tucked away some stuff in a technical appendix, but about electron spins he still says: “An electron has a spin of one-half, as does an up quark. Why this is possible is an amusing quirk of quantum field theory, but delving into it would take us even further afield than the rest of this relatively technical appendix.”

    1. David Yerle Post author

      Well, if Feynman failed at it then most likely nothing can be done about it… though I keep thinking there has to be a way. The key has to be in explaining why relativity combined with quantum mechanics gives rise to spin. But the thing is, it doesn’t: the Klein-Gordon equation is perfectly valid too. So what needs to be done is explain why relativity “splits” the possible types of particles into two, without using any math.
      Yup, that sounds hard.

      1. elkement

        I believe you get into explaining symmetries – Sean Carroll’s approach is similar. But you need quite some pages to get to the point and I am not sure if the more abstract symmetries can really be conveyed. Carroll does not give a tangible explanation – but probably I should also read the book again looking specifically for that.

        But talking about symmetries is the conventional math-y way still… Lorentz invariance… looking at equations and “suddenly seeing” that those things in the (Dirac) equation have to be objects with more dimensions than “expected”?

        Sometimes I think: Is is probably possible to start explaining physics from such an abstract perspective only? What if we wouldn’t know anything about Newton’s law and equations of motions? Go straight to group theory and symmetries (without using technical terms)?

        I really enjoyed Graham Farmelo’s biography on Dirac – made me admire his achievements even more. But it leaves me with even less hope to convey those theories somewhat …

      2. Jean Louis Van Belle

        I wrote about this. The post with a lot of math is: http://readingfeynman.org/2014/10/16/amplitudes-and-statistics/
        The post with much less math is: http://readingfeynman.org/essentials/
        But the easiest thing is really to think as follows: think of two phase vectors A and B. So simple vectors that are rotating clockwise or counter-clockwise, with varying magnitudes or not. It doesn’t matter. Then depict A+B, A-B, -A+B and -A-B. These are four possibilities of ‘combining’ the two phase vectors. However, you’ll find – obviously – that A+B = -(A-B) and that A-B = -(-A+B). So A+B = -(A-B), and A-B = -(-A+B), differ from each other by a phase factor that’s equal to 180 degrees: they just post in opposite directions. Now, because we always take the absolute square – think of these phase vectors as amplitudes, or wavefunctions – we get the same physical result. That means there are only two logical ways to ‘combine’ phase vectors. In Nature we have both: one way of combining them – by throwing a minus sign in – is the ‘fermion’ way of behavior. The other – no minus sign, or applying a minus sign to both – is the ‘boson’ way of doing business. 🙂 Does that help?

        1. David Yerle Post author

          Thanks! It does help. It’s still not at the “makes perfect sense to a teenager” kind of level, but it certainly beats “and now the creation and annihilation operators anti-commute”…

  2. john zande

    Suggestion, for every 100 positions given to physics students in faculties around the world 1 position should be allotted to a creative writer; a poet, an artist, someone who may conceptualise the information in a different way. This person will be awarded a philosophy degree but it’ll be their job (ultimately) to communicate physics to the world.

  3. dovhenis

    I’m a physics ignoramus but maintain that commonsense is the best scientific approach; hence my suggestion of gravity as the monotheism of the universe.

    Now, to my gravity suggestion add the following two quotations, one from prevailing classical physics and one from my postings:

    Electrons are thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton.

    * Evolution Is The Quantum Mechanics Of Natural Selection.
    * Quantum mechanics are mechanisms, possible or probable or actual mechanisms of natural selection.
    * Life’s Evolution is the quantum mechanics of biology.
    * Every evolution, of all disciplines, is the quantum mechanics of the discipline’s natural selection.

    Commonsensically this may answer your question re Pauli’s exclusion principle. It implies that present physics has not yet advanced far enough to discern the answer, that the answer will be seen when particles physics unravels evolution’s paths all the way down to the gravitons…

    Dov Henis (comments from 22nd century)

    1. David Yerle Post author

      Hi Dov,
      I’m sorry but I have to disagree. I don’t believe common sense to be the best scientific approach: in fact, the history of modern science is the history of common sense taking a historical beating.
      I have seen your theory on my blog several times and not published it for several reasons. The main reason is that comments should be about the article being presented and, as a general rule, take up less space than said article. If one wishes to write an article on their theory of everything, they are more than welcome to do so in their own website.
      That said, my main criticism towards your idea is that it is too vague: you can’t just state your three postulates and go take a nap. If you want to be taken seriously, you need to be able to formulate it mathematically and to make specific, quantitative predictions about different outcomes, such as cross-sections in the LHC. Once you have done this, you need to publish your article (preferably using LaTeX and following the standard scientific paper format) in a peer-reviewed magazine, so that other scientists may see your findings and criticise them, even put them to the experimental test.
      Anything else is not science, but philosophy. Philosophy is very welcome, of course, but not much use when you’re trying to build the next particle accelerator.

    1. David Yerle Post author

      I checked this article before I wrote mine, but I wasn’t completely sold. It seemed to me like it cheats a bit: it oversimplifies so much that the explanation is not really one, but more like a mnemonic device. That is, her explanation for why electrons can’t be in the same quantum state has very little to do (close to nothing) with the actual reason. I like my popularization to be the “real deal” in the sense that I translate the math but I don’t take shortcuts.
      Thanks for trying though!
      Haven’t written for a while have you? I have the feeling that our original “group” has kind of faded in the air. Johannes, livelyskeptic (who now has a different name but hardly ever writes)… there has to be some kind of “blog half-life” or something like that. Even I am barely touching the thing. Oh well…

      1. bloggingisaresponsibility

        Ok, I should have figured you would have seen that. The challenge for me is that I keep trying to map these concepts onto sense experience, and words like “spin” don’t help because of their associations. Ironically, the article I’m working on is about the baggage of words. Which brings me to your next point…

        I did write an article (since “Is Math a Psychology?”) titled “What Does it Mean to Identify With?”. That was a little over a week ago, and I hope to release the article I’m working on sometime this week.

        Amusingly, I have many drafts that I keep editing, but I hold off on releasing them because I want to make sure they have some value. So it isn’t so much a lack of ideas that stays my hand, but a concern with a lack of applicable ideas.

        I’ve also been meaning to ask what happened to some of our fellow bloggers like Johannes and Lively. I haven’t heard from Livelyr in ages. I wonder how she’s doing?

        1. David Yerle Post author

          Well, I haven’t heard anything from Johannes for ages, but Lively started a new blog (pipteinteron I think was the name) and then stopped writing altogether. I think people just get bored or maybe just don’t get as much out of it as they used to. In my case, I’m just overworked.
          I didn’t realise you had a new article, sorry! You’re in my feedly and you should’ve shown up. Maybe I accidentally marked it as read… anyway, going over your blog to read it now.

  4. physicistbyheart

    Well I was looking for answer the same question that you have David. The internet brought me here. I was so satisfied to see your comments which carry a sense of admittance of not knowing answer to the question Why does the wavefunction of fermions have to antisymmentric.

    To add to the conversation, I think experimentally it has be proven that antisymmetric wavefucntion is the ground state wavefunction for many electron atom like, Helium.

    So in that case I now have a question why do we even have the need to exchange the particles. What is the use of exchanging particles. And to answer myself to this quesion, I think I would say that, we exchagnge particles so that we can distinguish them(not sure if correct).

    An article related to the above topic is given here in this link.

    Just check the paragrpah “Distinguishing between particles” and see if that gives you some kind of idea to answer the basic question.

    If the QM dictates the particles to indistiinguishable, then why do we get different wavefunctions(minus sign is obtained when particle are exchanged), while exchanging the particles.

    I am sorry if I have confused you or not made sense, pls let me know.

    1. David Yerle Post author

      Hi! Actually, that’s a very good question. In the case of bosons, in fact, swapping two particles has no effect on the wave-function whatsoever. For fermions the wave-function gets a minus sign when you exchange two, which of course then raises the question: if they were indistinguishable, why do we get a minus sign exchanging them?
      I’m not in the most clear-headed of spots today so I’ll have to think about this more. It could be that I am also making no sense at all…

  5. Alessandro Drudi

    Sorry for showing up so late at the party, but I read somewhere that Pauli himself stated he was unable to find an intuitive explanation to his principle. I find some irrational solace in the idea that neutron stars break this annoying behavior of fermions

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