Tag Archives: quantum mechanics

Quantum Worldview

I have always loved the kind of science fiction where you think about a world that is largely like our own, but in some fundamental way different. What if we all lived underwater, or if the force of gravity was lower, or if the sun were a weaker star? To me, that’s what the world of quantum physics is like: in a lot of ways it’s similar to our own world, in fact it’s the basis of our world! But it’s also crazy and strange. So what would it be like if we were quantum creatures, if we could actually see how everything around us is quantized?

Well first, there’s what it means to be quantum. A quantum of anything is a piece that can’t be subdivided any further, the smallest possible unit. But this implies a sort of graininess, where rather than a continuous stream of, say, light, we start to see at the small scale that light is actually composed of little chunks, quanta of light. Imagine being able to see how everything around you is made of discrete pieces, from light to sound to matter. When the sun came up, you’d see it getting lighter in jumps. When you turn up your music, you’d hear each step of higher volume. And as your hair grew, you’d see it lengthening in little blips.

And at the quantum scale, the wave nature of everything becomes indisputably clear. We normally think of waves as something that emerges from a lot of individual objects acting together, like the water molecules in the sea, or people in a crowd. But if you look at quanta, you actually find that those indivisible packets of light or sound or matter are also waves, waves in different fields of reality. That’s hard to get your head around, but think of it this way: as a quantum wave, if you passed right by a corner, you could actually bend around the back of it a little the way that ripples going around a rock in water do. Things like electrons and photons of light actually do this, so for example the pattern below is made by light going through a circular hole, and the wave diffraction is clearly visible.

Amazingly, as a wave, you could actually have a slight overlap with the person next to you. This gets at something that’s key to quantumness: the probabilistic nature of it. Say I thought you were nearby, and I wanted to measure your position somehow to see how close you were to me. I’d need to get a quantum of something else to interact with you, but because it would be a similar size to you, it would slightly change your position, or your speed. We don’t notice the recoil when sound waves bounce off us in the macroscale world, but if we were very small we would! So there is actually a built in uncertainty when dealing with quantum objects, but we can say there’s a probability that they are in one place or another. So as a quantum creature, you can think of yourself as a little wave of probability, that collapses to a point when measured but then expands out again after. When I’m not measuring you, where are you really? Well I can’t say physically, and this is why you can have a little wave overlap with your neighbour without violating the principle that you can’t both be in the same place at the same time.

And imagine that you’re next to a wall. As a wave you may have a little overlap of probability with that wall. And if the wall is thin enough, as a quantum object there is actually some chance that you’ll pass through the wall entirely! This is called quantum tunnelling, and actually it’s happening all the time in the electronics inside your phone. Modern microelectronics work in part because we can use effects from the quantum world in our own, larger world!

It’s difficult to imagine a world where everything happens in discrete chunks, where I can see myself as a wave, where I don’t know where I am until someone else interacts with me. But this is the world at the quantum scale, and it’s not science fiction!

Small World

It’s a New Year, and I went ahead and tried something new! This is a video version of some of the cool nanoscale and quantum things I’ve written about before, created in collaboration with director/editor Kevin Handy. I hope you enjoy it; I had a blast making it.

Magnets, How Do They Work?

Most of us have had some experience with magnetism, whether it’s finding north with a compass, posting a grocery list with refrigerator magnets, or playing with magnetic toys that snap satisfyingly together. And from those interactions we can glean some basic information about magnetism:

  1. Magnetic fields, even weak ones such as that of the earth, exert a magnetic force.
  2. Magnetic forces are experienced by some objects but not others.
  3. Magnets are cool.

But what causes magnetism in the first place? The answer is quantum mechanical in nature, and relates to an idea we discussed when we talked about the spin-statistics theorem. Recall that fundamental particles are indistinguishable, meaning that if we have two electrons and two available states, we cannot tell the difference between a system where electron 1 is in state 1 and electron 2 is in state 2, and a system where electron 1 is in state 2 and electron 2 is in state 1. In fact, we can’t even figure out which is electron 1 and which is electron 2; mathematically they are indistinguishable. This, of course, relies on the particles having identical physical properties such as charge and mass, but all electrons do. Another way of saying that particles are identical is to say that you can’t tell the difference if you swap two particles, which physicists call exchange symmetry. Mathematically, it becomes necessary to write equations for multiple particles so that the equation is not modified by exchanging the particles, and this adds a term due to the exchange interaction. We can think of the exchange interaction as another factor affecting the landscape of energy available to the physical system, much like gravity or electrical interactions are factors. Thus, how to minimize the energy of the system depends on the exchange interaction. And it turns out that for some materials, the exchange interaction term causes a system with aligned spins to be lower energy than a system where the spins are randomly oriented.

This is the origin of ferromagnetism, a property of objects that are permanently magnetized. Their spins align to minimize energy, creating a magnetic field that can interact with other magnetically sensitive objects. Some materials are susceptible to magnetism without being permanently magnetic, which usually occurs because small regions of the material magnetize but the regions never combine to create a material-wide magnetization. This is often the case with alloys that include iron or other magnetic materials, and the regions of magnetization are called domains. (Magnetic domains in a rare earth magnet are shown in a microscope image below.) So when you put a magnet on a refrigerator, you are placing a permanent ferromagnet on an alloy and aligning a small region of magnetic domains, creating a force that holds the two together. But the magnet can’t align the spins in a wooden cabinet door, because other terms besides the exchange interaction are more important in that material, so magnets won’t stick to wooden objects.

The polarity of magnets, which are usually described as having a north and a south pole, also stems from the aligned spins. Spins aligned in opposite directions, say one pointing north and one pointing south, are generating oppositely oriented magnetic fields. This is a very high energy configuration, as you know if you have ever tried to hold the south ends of two bar magnets together. But rather than one magnet reordering the other, the magnetic forces generated push the magnets apart.

You might be wondering, is there any way to change the magnetization of a ferromagnet? And actually, there is. While the exchange interaction is one factor controlling the configuration of many particles together, another factor is more mundane: temperature! At high temperatures, the high ordering of all the spins being aligned costs more and more energy to maintain, because there is plenty of thermal energy jostling the spins around. This is why sometimes people talk about ‘freezing in’ magnetism, because if you place some high-temperature materials with unaligned magnetic domains in a magnetic field and then let them cool, they will stay magnetized even after being removed from the magnetic field. And, conversely, if you melt a ferromagnet, it loses its magnetic alignment (but not its fundamental propensity to magnetize, which will return as soon as it cools back down!).

Compass needles are also ferromagnets, but the origin of the earth’s magnetic field is more complex, and ties into the use of magnetism in circuits. Next time!

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.

The Electron Cloud

There is a popular image of the atom that shows the nucleus as a collection of balls, with ball-like electrons following circular orbits around them. The parallels to our own solar system, to the orbits of the moon around the earth and the earth around the sun, strike a chord with most people, but the depiction is inaccurate. It is based on the ideas of several prominent early twentieth century physicists, developed after the discovery of the electron in 1897 showed that atoms were not the smallest building block of nature. There are two serious mistakes in this image, and the actual structure of the atom is a lot more interesting.

The first problem is that the proton itself is not an indivisible particle: it’s composed of three quarks, subatomic particles which were hypothesized in the early sixties and observed in experiments beginning in the late sixties. The same is true of the neutron: it’s also composed of three quarks, though the flavor composition is different than that of the proton. (“Flavor composition”? Yes, quarks come in different flavors.) So those giant balls in the nucleus are actually comprised of smaller particles. At this point we believe quarks to be themselves indivisible, not composed of another even smaller particle.

But the second reason this picture is incorrect is that the electron doesn’t follow a linear orbit around the proton, the way gravitationally orbiting bodies do. In fact, due to the small mass of the nucleus and the even smaller mass of the electron, gravity is the least important force in an atom. The electromagnetic force, between the oppositely charged proton and electron, is much larger than the gravitational force between these tiny objects. But wait, you might say, if there’s such a large attractive force, shouldn’t the electron just spiral into the proton? This quandary illustrates perfectly why we can’t rely on classical physics, which was built up for objects comprised of billions of atoms, for the particles within a single atom. Because yes, if we had two oppositely charged billiard balls that have a weak gravitational interaction and a strong electromagnetic interaction, they will crash into each other! But, the electron is so small and so light that we cannot treat it as a classical object.

Here is where quantum mechanics come into play. Quantum mechanics as a whole is a set of mathematical constructions used to describe quantum objects, and it’s quite different than what’s used for classical, large-scale physics. There are all sorts of interesting consequences of quantum mechanics, such as the Heisenberg Uncertainty Principle, which states that for some pairs of variables, such as energy and time or position and momentum (mass times velocity), how precisely you can measure one depends on how precisely you are measuring the other. For each variable pair, there is a basic uncertainty in the measurement of both, which is very small but becomes relevant at the quantum scale. This shared minimum uncertainty between related variables is a fundamental property of nature. For momentum and position, this leads to that old joke about Heisenberg being pulled over for speeding: the police officer asks, “Do you know how fast you were going?” and Heisenberg responds, “No, but I know exactly where I am!”

What the uncertainty principle means here is that the electron is actually incapable of staying in the nucleus. Imagine a moment in time where the electron is within the nucleus: now its position is very well known, so there is a large uncertainty in its momentum. Thus the velocity may be quite high, which means that a moment later the electron will have moved far from the nucleus. In fact, because of the uncertainty in position, we cannot ever really say where in space the electron is. It is more accurate to talk about its position as determined by a probability cloud, which is denser in places that the electron is more likely to be (near the nucleus) and less dense where it is less likely to be (far from the nucleus). This also takes into account the wave nature of the electron as a quantum object, which we’ll get into another time.

With this knowledge, we can discard that old image of an electron orbiting a nucleus. A single electron, even though it is measurable as an individual, indivisible particle, exists as a cloud around the nucleus. The shape of the cloud is described by quantum mechanics, and as we add more electrons to the atom, we will find a whole gallery of electron cloud shapes. These shapes are the heart of interatomic bonding, as we will see.