Sameness leaves us in peace
but it is contradiction that makes us productive.
~Johann Wolfgang Von Goethe

In previous entries, I’ve discussed the wave nature of particles and some consequences of that wave nature, how electrons occupy specific energy states in atoms, and how particles obey the laws of probability. This is all pretty weird stuff. However, there’s another strange phenomenon in quantum mechanics that I haven’t discussed. That phenomenon is the Pauli exclusion principle.

The Mystery of Stability

An atom is made of protons, neutrons, and electrons. A good (but not quite right) model of the atom is that the negatively charged electrons orbit around the positively charged nucleus, which is made of protons and neutrons. In the early 1900s, chemists discovered something strange: atoms with an odd number of electrons are much more like to bond with other atoms and form molecules than atoms with an even number of electrons. 

As we learned from Niels Bohr, electrons in an atom occupy certain special “energy levels,” each one a different distance from the nucleus. Chemists wondered if perhaps the number of electrons in each level had something to do with how reactive an atom was. In 1919, Irving Langmuir suggested that each energy level could only hold a maximum of two electrons. Electrons “prefer” to be in the lowest possible energy level—they even give up energy in the form of light to ensure this. Langmuir’s suggestion means that, as electrons are added, the low-energy orbits must fill up first. Each orbit gets only two electrons before it’s full, which forces electrons that come later to occupy higher and higher energy levels.

Langmuir argued that an atom was in its most favorable (i.e., lowest-energy) state when all of its occupied orbitals were full. This meant that the outermost orbital needed to have two electrons in it. If two atoms had half-full orbitals, each could donate its outermost electron to form a joint orbital, which was more “desirable” for the atom. We now call this process covalent bonding.

But why was each orbital restricted to just two electrons? Langmuir didn’t have an answer to this question. He simply asserted the law of two electrons per orbital because it matched experiment. This mystery wouldn’t be solved for another five years.

Enter Pauli

In 1924, Wolfgang Pauli suggested that atomic orbitals admitted only two electrons because of a special property of electrons. He proposed that two electrons cannot occupy the same “quantum-mechanical state.” In other words, they can’t be in the same place and have the same energy and momentum. (Remember that, since electrons orbiting an atom behave as waves, the electrons fill the entire orbital. As a result, two electrons occupying the same orbital always occupy the same space.) This idea is called the Pauli exclusion principle.

But if no two electrons can occupy the same state, why can two share an atomic orbital? The answer, Pauli argued, must be that they are not in the same state. Pauli (correctly) guessed that electrons also have another quantum-mechanical property, one we now call “spin.”

The Spin Connection

Intuitively, we think of the spin of an electron as the rotation of an electron around its central axis—like how the Earth spins to create day and night. Because an electron is charged, this rotation generates a magnetic field. Intuitively, this works the same way as an electromagnet. However, there are two reasons why any intuitive understanding of spin must be wrong.

Firstly, as far as we know, electrons have no physical volume. An electron cannot spin around its central axis because there’s nothing to spin—it is a zero-dimensional point. The way we know that the Earth is spinning is that we can tell that some point at the surface moves around some other point. (For example, we can say that Tokyo, the Amazon rainforest, your grandma’s house, Ayers Rock, or anywhere else in the world moves in a circle around both the North and South Poles. We can also therefore say that the poles don’t move relative to each other.) However, because an electron is just one point, we can’t talk about one point moving or not moving around another. The entire concepts of revolving, axes, or revolving around axes doesn’t even make sense.

Secondly, the axis of rotation for a non-quantum-mechanical object can point any way it likes. The Earth could rotate around an axis ninety degrees from the current one, so that the North Pole always faces the sun and the South Pole always faces away from it. (This actually happens on Uranus, which has an axial tilt of almost one hundred degrees off of the plane of the rest of the solar system.) However, spin can’t possibly behave like this. The number of possible directions in which an electron can spin must be equal to the number of electrons that can fit into an orbital. Otherwise, each new electron could spin in a different direction and the Pauli exclusion principle would allow any number of electrons in a single orbital. Therefore, electrons must have a limited number of possible spin directions. And since we know from Langmuir that only two electrons can occupy a given orbital, electrons must have precisely two possible spin directions.

Since then, we have experimentally verified Pauli’s exclusion principle. Electrons do, in fact, have only one axis of rotation, around which they can spin either clockwise or counter-clockwise. We call these states “spin up” and “spin down.” Because spin states are waves, an electron can also be in a “superposition state” where it is both spin up and spin down.

If you don’t understand all this, don’t worry–no one else does, either. Although physicists think of spin as an electron spinning around a central axis, our analogy is oversimplified and breaks down quickly. Spin is a purely quantum-mechanical phenomenon, with no analogue in the macroscopic human world. Our brains are simply not built to understand it.

The Exclusion Principle and You

Thanks to Pauli’s exclusion principle, Langmuir’s work was completed and the mystery of atomic stability was solved. Together, the exclusion principle and the existence of two spin states explain why only two electrons can exist in an atomic orbital. But where else does the Pauli exclusion principle show up? Is it relevant? Does it exist? Of course the answer is “yes,” or I wouldn’t have asked.

The Pauli exclusion principle is important in every situation where electrons are important. As I will discuss next time, the exclusion principle plays an important role in how some materials conduct electricity and why others don’t. This understanding opened the door to the development of the transistor and modern computers. More existentially, the Pauli exclusion principle means that electrons cannot occupy the same space, which prevents physical things from overlapping. This is why we can build structures (and exist as terrestrial organisms) without being afraid that everything will fall through the ground and into the Earth’s core. Honestly, it’s why anything, the Earth’s core included, can exist as a bounded object separate from anything else. (On the other hand, we can’t walk through walls. Too bad.)

Electrons aren’t the only objects that obey the Pauli exclusion principle. There is an entire class of particles, called “fermions,” that obey this property. An interesting celestial phenomenon results from the fact that neutrons are fermions. When a star of the right size and the right composition collapses after it runs out of fuel, it becomes a neutron star. The neutrons in the star obey the Pauli exclusion principle, so they can’t occupy the same space. This prevents the star’s complete collapse and acts as a “Pauli pressure” holding the star a hair’s width away from becoming a black hole.

The exclusion principle also explains why some materials can form a Bose-Einstein condensate and why some can’t—fermions can’t get too close to each other and thus can’t form the condensate. It also explains why different isotopes of helium become superfluids at different temperatures. Helium-3 is a fermion, so its atoms repel each other strongly; it refuses to form a superfluid until it gets so cold that its atoms merge together and cease to be fermions. Helium-4, though, is already a boson (a non-fermion), so it can become a superfluid at a much higher temperature than Helium-3. (Here I’m glossing over another detail of spin, called the spin statistics theorem. If you’re interested, let me know in the comments and I’ll discuss it another time.)

Further Reading

There’s a wealth of technical information about the Pauli exclusion principle and spin. I’ve tried to find the most non-technical references.

• Encyclopedia Britannica on the Pauli exclusion principle.
• Brian Cox has some fascinating things to say on the Pauli principle.
• Sean Carroll has some equally fascinating disagreements with Brian Cox.
• Physics Intuitions (a blog I may have to start reading!) tries to frame quantum spin intuitively. A little is lost in translation, but overall, this is a very elegant explanation.

Questions? Comments? Hatemail? Volunteers?

As always, if you have any questions, comments, corrections, or insults, please don’t hesitate to let us know in the comments!

I also have a special request this week. If any of you readers are astronomers or astrophysicists, would you be interested in writing a guest post on neutron stars? It’s not my area of expertise and I’m sure I didn’t cover all the interesting things about them. (In fact, I probably got something wrong.)