Tag Archives: pauli exclusion principle

Spin, Rotation, and a Plate Trick

In my introduction to the quantum number spin, I mentioned that particles can have half-integer or integer spin, and that which they have deeply affects their behavior. This is not an easy statement to understand, especially without seeing the math. The allowed values for spin come from solutions to quantum mechanical energy equations. But what do differences in these values mean? How does a spin-1/2 particle behave differently than a spin-0 particle?

One major difference is in the behavior under rotation. When we try to calculate how rotation affects a particle with spin-0, we find that it doesn’t matter: the particle is indistinguishable before and after any rotation. However, a spin-1 particle requires a 360° rotation to return to its initial state, and a spin-2 particle requires a 180° rotation to return to its initial state. This may seem strange, but what it means is that the spin value describes the symmetry of the particle. If you imagine a deck of cards, the spin-2 particles are like face cards that look the same when rotated 180°. Spin-1 particles are like number cards which must be rotated 360° to look the same as they did when they started. Particles with integer spin are called bosons, after the Indian physicist Satyendra Bose.

There are no playing cards which must be rotated 720° in order to look the same, and yet this is the case with spin-1/2 particles. There are few macroscopic objects that can demonstrate this property, but one of them is your hand! Place any object on your hand, palm up, and rotate it without dropping your palm. After 360° you will find your arm to be pretty contorted, but after 720° of rotation your arm has regained its initial position! Another way to think of it is that, instead of a 360° rotation bringing the object back to its initial state, which would be like multiplying by 1, the 360° brings the object to another state like multiplying by -1, and then an additional 360° rotation multiplies by (-1)*(-1) which equals 1. Every spin-1/2 particle shares this behavior, such as quarks (the constituents of protons and neutrons) and electrons. We call these particles fermions, after the physicist Enrico Fermi.

That factor of -1 becomes important because of the idea in quantum mechanics that particles are interchangeable or identical. That is, we cannot tell one specific electron from another. Mathematically, you can state this by writing a function that describes the positions of two particles, and seeing what happens to that function when you exchange the particles. If you do this, what you find is that bosons are symmetric under particle interchange and the function stays the same, but fermions are antisymmetric under particle exchange, and the function is multiplied by -1.

This idea, that bosons are symmetric and fermions are antisymmetric under exchange of identical particles, is called the spin-statistics theorem. A thorough proof requires relativity and quantum field theory, but the fundamental cause is the differing rotational behavior due to spin as a measure of symmetry. One very important consequence of all of this is that if you have two fermions occupying the same state, and you exchange them, you find that the function describing their position cancels out to zero. This is a mathematical statement of the Pauli exclusion principle forbidding two fermions from being in the same quantum mechanical state!

On the other hand, we find that bosons are perfectly happy to all pile into the same quantum mechanical state, at least at low temperatures. This is the concept behind the Bose-Einstein condensate, the state of matter experimentally realized only 30 years ago in which bosons can be cooled into occupying the same state.

I hope this makes the connections between spin, the Pauli exclusion principle, and particle types clearer. But if nothing else, the rotational exercise with an object on your hand, better known as Feynman’s plate trick, is fun at parties.

What is spin?

First the basics: spin is an intrinsic property of matter, like charge or mass. It is measurable in the real world by observing interactions with magnetism, and is the basis of technologies like MRI and hard disk drives!

We of course recognize the verb ‘to spin’, which means to rotate around a fixed axis the way that wheels, figure skaters, and the Earth do. But the word spin is also used to describe a fundamental property of particles. We have already talked a little about a fundamental property, charge, which was useful because a lot of the important forces at the atomic scale are electromagnetic and thus related to charge. And we remember that mass, another fundamental property, determines how matter interacts via the gravitational force. Spin is a bit different.

The idea of particles having an intrinsic spin first arose during the development of quantum mechanics, when Wolfgang Pauli and others noticed that part of the mathematical solution for particle states resembled angular motion, as if the particles were physically spinning around an axis. But unlike spinning at the macroscopic scale, quantum spin can only occur at a few discrete values: integer and half-integer multiples of ħ, the reduced Planck constant. The allowed values of spin are clustered around zero, and the ħ factor is dropped by convention because particle physicists like to make things look simple. So a photon, the quantum of light, has spin 0, whereas electrons and quarks, which make up protons and neutrons, have spin 1/2. There are also particles with spin 1, 3/2, and 2. As with charge, spin is reminiscent of a behavior we see in the macroscopic world, but its values are quantized into a few allowed values.

Spin can have one of two polarities, meaning we can have an electron with spin +1/2 and one with spin -1/2. And charged particles like the electron actually respond to magnetic fields differently if they have positive or negative spin! This is because the motion of a charged particle creates a small magnetic moment, which will be aligned in one direction for positive spin and the opposite direction for negative spin. This is the basis of the famous Stern-Gerlach experiment, in which atoms with one free electron are sorted by their spin under the influence of a magnetic field. But it’s also the basis of nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI), two related techniques for determining the composition and structure of either chemical substances or human patients! Strong magnetic fields can be used to align spins within any object, and how quickly the spins decay back to their original orientation gives information about what is inside the object. Currently, researchers are trying to build circuits that use spin instead of charge to carry information, which is called ‘spintronics’.

But at a more basic level, when we talked about chemical bonds we skipped over the importance of spin. The reason spin matters for bonding is due to the Pauli exclusion principle, the idea that no two electrons can share the same quantum state. In the development of quantum mechanics, it became clear from the data that even if all the available energy states were mathematically accounted for, there still seemed to be a degeneracy in which two electrons shared what was thought to be the same quantum state. This can be explained with a new quantum number, which we call spin. So spin is another factor of the electron cloud shape and is critical in the understanding of chemical bonding.

But there are actually even more strange things about spin than I can fit in this post, including the fact that the Pauli exclusion principle only applies to particles with half-integer spin! Half-integer and whole-integer spin particles are fundamentally different from each other, in some pretty interesting ways, but why is a story for another time!

Electrons, Bonding, and the Periodic Table

The structure of the periodic table of elements is a bit weird the first time you see it, like a castle or a cake. If we just read the periodic table top to bottom and left to right, we are reading off the elements in order of increasing number of protons. However, if this were the only useful ordering on the periodic table, it could be a simple list. The vertically aligned groups on the periodic table actually represent the chemical properties of the elements. Dmitri Mendeleev developed the table in 1869 as a way to both tabulate existing empirical results, and predict what unexplored chemical reactions or undiscovered elements might be possible. It was revolutionary as a scientific tool, but the mechanism behind the periodicity was not understood until decades later. As it turns out, the periodicity of chemical behavior corresponds to the bonding type of the outer electrons in different atoms.

To understand what that means, we can start by looking at the elements on the left side of the periodic table. Hydrogen has only one proton, so the electrically neutral form of hydrogen has only one electron. This single electron is a point particle, jumping around the nucleus. The electron exists in a probability cloud, whose shape is given by the lowest energy solution to the quantum mechanical equations describing the system. These quantum states can be distinguished by differing quantum numbers for various quantities like spin and angular momentum, and we will talk about these in more depth later on. When we add additional electrons, they all want to be in the lowest energy state as well. Sadly for electrons but happily for us, no two electrons are able to occupy the same quantum state: they must differ in at least one quantum number. This is known as the Pauli exclusion principle, and was devised to explain experimental results in the early years of quantum mechanics. So while the single electron in hydrogen gets to be in the lowest energy state available for an electron in that atom, in an atom like oxygen, its eight electrons occupy the eight lowest energy states, as if they are stones stacked in a bucket.

But what’s really interesting about these higher energy electron states is that they have different shapes, as we can see by the mathematical forms that describe the possible probability distributions for electrons. So while the electron cloud in a hydrogen atom is a sphere, there are electron clouds for other atoms that are shaped like dumbbells, spheres cut in two, alternating spherical shells, and lots of other shapes.

The electron cloud shape becomes important because two atoms near each other may be able to minimize their overall energy via electron interactions: in some configurations the sharing of one, two, more, or even a partial number of electrons is energetically preferred, whereas in other configurations sharing electrons is not favorable. This electron sharing, which changes the shape of the electron cloud and affects the chemical reactivity of the atoms involved, is what’s called chemical bonding. When atoms are connected by a chemical bond, there is an energy cost necessary to separate them. But how atoms interact depends fundamentally on the shape of the electron cloud, determining when atoms can or can’t bond to each other. So the periodic table, which was originally developed to group atoms with similar chemical properties and bonding behaviors, actually also groups atoms by the number and arrangement of electrons.

Now, there is a lot more that can be said about bonding. You can talk about the inherent spin of electrons, which is important in bonding and atomic orbital filling, or you can talk about the idea of filled electron shells which make some atoms stable and others reactive, or you can talk about the many kinds of chemical bonds. It’s a very deep topic, and this is just the beginning!

Since every real world object is a collection of bonded atoms, the properties of the things we interact with, and what materials are even able to exist in our world, depend on the shape of the electron cloud. Imagine if the Pauli exclusion principle were not true, and all the electrons in an atom could sit together in the lowest energy state. This would make every electron cloud the same shape, which would remove the incredible variety of chemical bonds in our world, homogenizing material properties. Chemistry would be a lot easier to learn but a lot less interesting, and atomic physics would be completely solved. Stars, planets, and life as we know it might not exist at all.