A Quick Introduction to Quantum Mechanics

You probably already have some notion of what quantum mechanics is: it describes things at a very small scale and it says that nature obeys weird and counterintuitive rules. Here we’re going to give a brief rundown of the basics of quantum mechanics, introducing it in a way which is non-mathematical but mirrors how physicists like to think about it. This means we will introduce some important concepts, such as that of a quantum state, before we describe the weird quantum phenomena it is responsible for. If you’ve never encountered the topic before, it may well be worth reading through this page twice before moving on.

Measurement

First let’s discuss the scope of quantum physics. As you may know it doesn’t directly describe the familiar, everyday physics of the world around us. Physicists have a special term for this type of physics, which describes why balls fall to the ground due to gravity, how fluids swirl around containers or down rivers, and how light propagates as an electromagnetic wave: classical physics. A detailed course on classical physics is beyond the scope of this site—we’re only concerned with how it differs from quantum physics. Our first key point is:

Apart from the different scales on which we apply quantum and classical physics, another important difference is what it means to measure a classical versus a quantum system. You probably make classical measurements all the time: when you stand on the bathroom scales you measure your mass, and when you use a tape measure you’re measuring the distance between two points. This brings us to our next key point, something you probably never consider when making such measurements:

This might seem blindingly obvious, but it’s important to establish it now as it turns out making measurements in quantum mechanics is very different. Let’s now establish one other thing about a classical system. System can mean basically anything we’re attempting to describe with classical mechanics—a nice simple example is a ball rolling along a flat surface.

Quantum States

We say that the information about a quantum system is contained in a quantum state, just as we do above for a classical system and a classical state. When we take measurements of classical system, we’re accessing information contained in the classical state which describes it. This is also true of making measurements on a quantum system. This is where there is a key difference—a distinction so important that if you understand this you’re well on your way to having a better understanding of quantum mechanics than most people do.

This is a really weird idea—something which is always true of a classical system is never true for a quantum one. It also means that a quantum state is a physical thing, something more than just a set of measurements. It also leads us to ask what result we do get when we measure a quantum system?

The way quantum states behave when measured can help us to understand another really important thing about quantum systems: what the term quantum actually means.

Now we’ve outlined the basic principles of quantum mechanics, we will show how it is described mathematically. For a more complete discussion of the mathematics, see reference [1] at the bottom of the page.

Mathematical Description of Quantum Mechanics

Before discussing the mathematics of quantum mechanics, let us first think about how the classical world is described mathematically. Usually, the approach is to write down an equation which, when solved, yields a mathematical expression or a set of expressions which contain all the information about a physical system. Say we have a pendulum and we want to know its velocity, position and acceleration at any point in time. With some assumptions about the forces involved we can write down an equation, known as the equation of motion, for the system which relates the forces to the acceleration of pendulum—note that this equation does not tell us what the acceleration actually is, it just tells us how the acceleration is related to the forces that act on the pendulum. It’s called the equation of motion of the system. Solving the equation of motion yields a mathematical expression which tells us the position of the pendulum at any given time.

Now that we’ve discussed the mathematical description of a pendulum, let us take the leap to a quantum system. Here it is not so simple as to find an equation relating acceleration to force. However we do need to start with an equation that can be solved to obtain information about the quantum system. Fortunately this equation does exist and is known as the Schrödinger equation.

We have barely scratched the surface of the theory of quantum mechanics. There is much more to explore such as the connection of the wavefunction to actual waves, the famous Heisenberg uncertainty principle, quantum entanglement and what we mean by operators.

1. Leonard Susskind, The Theoretical Minimum Lectures, https://theoreticalminimum.com/courses/quantum-mechanics/2012/winter

Emergence

As far as we know, all the matter in the universe that we have any detailed knowledge of, including our minds and bodies, is controlled by the same set of fundamental laws. However, in the words of physicist Philip Warren Anderson, “The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe.” It might seem like this is exactly what physicists are trying to do, but starting from the most fundamental laws is not always useful. The physics of many (many means approximately 10²³) atoms is different to the physics of a single atom, which is what Anderson argues in his paper ‘More is Different’ [1]. It’s not just that there are more atoms, but there are phenomena we observe when many atoms come together, for example to form a crystal, that we could not possibly observe with a single atom.

Think about the air around you, which is made up of molecules, which are made of atoms, and we can go all the way down to electrons and quarks, which as far as we know are fundamental particles. The physics of quarks and electrons is very well understood, so we could try to write down all the relevant equations for all the air molecules in a room. One might think that, in principle, we could calculate the motion of every single air molecule. There are roughly 10²³ air molecules in a room and this is not even the biggest number that will be dealt with in the computation of such a problem. Because we would have to consider the interaction of every molecule with every other molecule, whatever computer—and it would have to be a computer at this point—we use would have to deal with a number larger than 10¹⁰²⁴ which vastly exceeds the number of particles in the observable universe [2]. This is obviously a physical limitation.

The behaviour of a large number of fundamental particles with complex interactions between them cannot simply be understood by building up from the fundamental laws that govern each particle. New properties often emerge and these emergent phenomena can be found everywhere in condensed matter physics. For example, thermal properties of solids are understood in terms of ‘fictitious’ particles called phonons. Phonons are particles that emerge from the vibrations of the crystal lattice (which makes up a solid). We really can ‘see’ these phonons when we probe materials, meaning we can bounce neutrons off them and measure their energy-momentum relationship, just like any other particle. Phonons and other emergent particles (often known as quasiparticles), however, will never be found at the Large Hadron Collider, which looks for fundamental particles. They only arise as a consequence of the complex interaction of many particles.

John Conway’s Game of Life is a really fun, visual way to see complex behaviour emerging from simple rules. Although the fundamental laws of physics aren’t simple in the same way that the rules of the Game of Life are, we can still draw parallels between what happens in the game, and what happens in our universe. The game is simple: a square grid of cells is defined and each cell can either be alive or dead (usually coloured black or white respectively). The game begins with an initial configuration of cells, and a new configuration is obtained at each time-step based only on the previous configuration.

There is nothing more than these simple rules, yet we see rich behaviour emerging such as the ‘queen bee shuttle.’ See if you can make it yourself in the interactive game board below. Of course the Game of Life is different in that we can actually compute all the possibilities from the fundamental rules but it allows us to see emergent behaviour that doesn’t need to be described in terms of the fundamental rules.

As well as new properties emerging, systems of many particles often do not possess the same symmetries as their constituents—they are different in that they break the symmetry which their constituents hold. We say that a system has a symmetry if the system is unchanged before and after a transformation associated with that symmetry. For example, if a system has rotational symmetry, rotating the system by some angle does not change it: if no one told us whether a rotation had been applied or not we would not be able to tell. There are more abstract symmetries such as time reversal symmetry, or parity but the general idea of leaving the system unchanged is still there. Many materials have been found not to possess all of these symmetries which the fundamental particles that make up the material do possess. See our article on topological classification here to find out how physicists make use of this idea.

1. Philip Warren Anderson, More is Different, available at Science: http://science.sciencemag.org/content/177/4047/393
2. Stephen Blundell, Emergence, causation and storytelling: condensed matter physics and the limitations of the human mind, available on arXiv: https://arxiv.org/abs/1604.06845

Spin and the Pauli Exclusion Principle

This article will be a discussion of some of the key properties of elementary particles important to condensed matter physics. First of all, what do we mean by elementary particles? These are particles that, as far as we know, have no sub-structure, so the electron is an elementary particle but an atom is not (because it is made up of protons, electrons and neutrons). In fact, protons and neutrons are not elementary particles themselves because they are made up of elementary particles known as quarks [1]. All elementary particles can be classified based on properties called mass, charge and spin. The property of mass should already be familiar from day-to-day life—objects with more mass feel heavier when you pick them up. The charge of a particle controls how strongly it interacts with electrical fields. Here we will discuss spin.

Spin

The spin of a particle is defined as the intrinsic angular momentum of the particle. If this doesn’t mean anything to you, that’s okay. What we will be most interested in is how the electron spin affects its behaviour. For charged particles, like the electron, spin allows them to interact with magnetic fields through their magnetic moment associated with spin. A magnetic moment is a quantity that represents the magnetic strength and orientation of an object that produces a magnetic field. A familiar example of this would be a bar magnet—we can directly visualise the magnetic field with iron filings. Both elementary and composite particles can have spin. The particle does not actually spin, in fact the notion of a solid particle is a bit fuzzy, but its behaviour in a magnetic field is similar to that of a spinning charged classical object which is why we use this term. When we measure the spin of a particle, there are two things we can measure: we can measure the magnitude of the spin and the projection of its spin in a chosen direction.

To talk about the projection of spin, the following classical picture might be useful (but ultimately spin is like nothing in the classical world).The figure below shows a classical object rotating—a spinning top. Here, we can measure the projection of its axis of rotation on to a chosen direction, which is labelled z. The angle between the z direction and the axis of rotation indicates how much we would have to tilt the spinning top for its axis of rotation to be the same as the the z direction. This is roughly what the term projection means, but there are some more details involved in the proper definition. To measure the projection of a particle’s spin, we choose a direction in space, which will usually be the direction of an applied magnetic field, and we obtain a number through experiment which is interpreted as the projection of the particle’s spin in that direction—this is a little bit like the angle in the classical picture. The number that we obtain is quantised, which means that when we measure it we will always find it to be a certain discrete number, a very different situation to what happens in the classical case where the angle can vary continuously. For a spin-half particle for example, we will only ever measure values +½ or -½ for the projection of its spin. The +½ value corresponds to the spin ‘pointing’ in our chosen direction and the -½ value corresponds the the spin ‘pointing’ exactly opposite to our chosen direction. We call these two situations spin-up and spin-down respectively.

The magnitude of spin is a bit like the magnitude of the angular momentum of a spinning object. Even though the rigorous definition of spin is that it is the intrinsic angular momentum of a particle there is no physical rotation of the particle. The numbers we get when we measure the magnitude and projection of spin are known as quantum numbers .

Elementary particles can be split into two classes based on the quantum number giving the magnitude of the spin, which can be only an integer (1, 2, 3, 4, …) or a half-integer (1/2, 3/2, 5/2, 7/2, …). Particles with integer spin are known as bosons and particles with half-integer spin are known as fermions. Fermions obey something called the Pauli exclusion principle.

Identical particles and the Pauli exclusion principle

The Pauli exclusion principle states that two or more identical fermions cannot occupy the same quantum state within a quantum system at the same time. Here we mean identical in the sense that there is no possible experiment that can distinguish between so called identical particles. This sounds obvious, but think about collisions between balls on a pool table and let’s say all these balls have identical mass, shape, are made of the same material and are even the same colour. You would probably call these balls identical but they are not in the quantum mechanical sense because there is an experiment that can distinguish between them. We can film the collisions on the pool table, and in principle we could label and keep track of every single ball, using video editing software maybe, which provides a way for us to distinguish between them. There is no such experiment we can perform for a quantum mechanical system. Quantum mechanical pool balls would be completely indistinguishable.

Most of the matter around us is governed by the Pauli exclusion principle because electrons are spin-half particles, and we usually describe the behaviour and properties of a material in terms of the behaviour of the electrons within it. An electron in an atom has a number of quantum states it can occupy. When there are multiple electrons in an atom, they must obey the Pauli exclusion principle, which means that each available quantum state can only have one electron at a time occupying it. It’s a bit like how there is only ever one person at a time in each seat on a bus (usually). Unlike the bus however, the electrons will obey a strict order when filling the the quantum states in an atom. This leads to an elaborate structure of orbitals, shells and subshells, which determines the chemical properties of a given atom and the way that it interacts with other atoms and compounds. In fact, many of the phenomena that we observe in materials are consequences of the quantum mechanical behaviour of electrons.

1. CERN website, the Standard Model, https://home.cern/about/physics/standard-model

Metals, Insulators, and Semiconductors

Materials are broadly split into three groups according to their electrical conductivity: metals, insulators, and semiconductors. Electrical conductivity is a measure of how easy it is for a material to conduct electricity: metals, such as copper, have a high conductivity and insulators, for example rubber, have a low conductivity. The electrical conductivity of semiconductors lies somewhere in-between. This classification is based on how electrons within atoms organise themselves when many atoms come together to form a solid.

There are a number of quantum states available to all the electrons within an atom. Each quantum state can only be occupied by one electron at a time (due to the Pauli exclusion principle). For an isolated atom, each quantum state is specified by four numbers. The first is the energy an electron has while it occupies said quantum state—the other three are not so important for now. One of the strange predictions of quantum mechanics, which we know to be true, is that this energy number can only take certain values. It’s a bit like having a car that can precisely travel at speeds that are multiples of 10, say, but no other speeds are accessible. It would be perfectly possible for this car to travel at 30 mph but 34 mph or 45 mph, for example, would be absolutely impossible. This really is what happens in the quantum world—there are certain energies which are forbidden for all the electrons within an atom.

Without considering the detail of the other three numbers that specify a quantum state, the electrons in an isolated atom are organised in discrete energy levels. There are a number of electrons at each value of energy (the other three numbers make sure that electrons with the same energy are indeed in different quantum states), with no electrons having energies with values that lie between the allowed energies. When a large number of atoms come together to form a material, they interact in a way that results in our material only having two energy levels that its electrons can occupy—sort of. We actually call each of these ‘energy levels’ a band because within each band there is actually a range of energies available to the electrons, not just a single energy, but the energies that lie between the bands are forbidden. The lower energy band is known as the valence band  and the higher energy band is known as the conduction band.

In order for a material to conduct electricity, the electrons within said material have to be free to move around. It turns out that this requires empty quantum states that the electrons can easily access (there may be empty quantum states that the electrons can’t access easily). In a conducting material, there is actually no gap of forbidden energies, so there is effectively only one band which isn’t completely filled up with electrons, which means that the electrons are free to move around. Electrons being able to move around is exactly what is meant by saying a material conducts electricity. In an insulating material, the bottom band (the valence band) has all its quantum states filled up with electrons and the gap of forbidden energies is too big for the electrons to easily access the empty quantum states in the upper band (the conduction band). Semiconductors have the same structure as insulators, but have a much smaller gap. The empty quantum states in the conduction band can be made available to the electrons in the filled valence band by adding impurities to the semiconductor or increasing the temperature.

The understanding of semiconductors through quantum condensed matter physics has led to perhaps the greatest technological advance of the modern era: the development of semiconductor devices. One device in particular, the transistor, forms the basis of every modern electronic circuit and every phone, tablet and computer literally contains billions of transistors. The invention of the transistor is credited to a team from Bell Labs in 1947: John Bardeen, Walter Brattain, and William Shockley [1]. The three were awarded the Nobel prize in Physics 1956 for their research on semiconductors and discovery of the transistor. We may not be too far from another technological advance as recent work connecting quantum condensed matter theory with topology may help overcome one of the biggest challenges in the development of quantum computers.

1. Steven H. Simon, The Oxford Solid State Basics

Superconductors and Superfluids

Superconductors are some of the best understood quantum materials, and are starting to find applications in technology. At room temperature, all known materials have some electrical resistance. This ranges from metals with a small resistance, which we call conductors (copper, gold, aluminium and iron are some of the best, which is why they’re used in electronics), through to materials with a very high resistance which we call insulators (plastics, for example, are used to insulate electrical wires for our safety). Materials which are superconductors also have some electrical resistance at room temperature. But when superconductors are cooled down to very cold temperatures, something remarkable happens—they completely lose all electrical resistance (i.e. their resistance drops to zero).

This phenomenon is an example of a phase transition. Elsewhere on the website we discuss how physicists describe the normal (non-superconducting) and superconducting phases of these materials. But there exists another theory, developed by Bardeen, Cooper, and Schrieffer (BCS) that describes superconductivity in terms of what happens to the particles that carry electricity through the material (electrons).

What’s important to understand is that in conductors, electrons carry electricity by moving around, and as they do so they interact with the lattice (a repeating, 3D grid) made up of the atoms of the material. This means that the energies of the electrons are affected by the nature of the lattice (this is the basis of how we understand metals and insulators); but it also means that the electrons can affect how the atoms of the lattice vibrate. In fact, at normal temperatures it is these lattice vibrations which scatter electrons and give the material its resistance. But at low temperatures they have a very different effect. The lattice vibrations and the electrons interact in such a way that the electrons will pair up and move through the material without any resistance. These are called Cooper pairs, and are responsible for superconductivity. If the temperature of a superconductor is raised, the electrons have too much energy to stay paired up and break apart instead, returning the material to its normal state with electrical resistance. The BCS theory, proposed in 1957, won its creators the 1972 Nobel Prize.

One familiar application of superconductors today is in MRI scanners. The principle of their operation is that in a very large magnetic field, hydrogen atoms in water (the H in H₂O) will resonate in a way that produces detectable radiation. This allows the human body to be imaged. The MRI ‘tube’ contains a large electromagnet, a coil of wire which produces a magnetic field inside when an electrical current is passed through it. The higher the current the larger the field. If the coil was made from a normal metal—even a good conductor like copper—only so much current could be passed through it before it would become too hot (this is called resistive heating, which is how kettles work, and is why your laptop warms up as you use it) and the coil would melt. Superconductors, having no resistance, avoid this problem, meaning very high currents can be passed through it. This generates magnetic fields 50000 times stronger than the Earth’s natural magnetic field in the case of MRI scanners. And the only price to pay is that the superconductor has to be kept very cold at 4K (-269°C).

Superfluidity

Superfluids are materials closely related to superconductors. Superconductivity is the flow of charged particles (electrons) through a material without resistance; superfluidity is the flow of particles without viscosity. The technical meaning of viscosity is the same as its day-to-day usage except it extends to gases as well as liquids—treacle is very viscous, whereas air isn’t (when physicists talk about fluids we mean both gases and liquids). A superfluid has no viscosity at all. Like in superconductors, this only occurs in a very low temperature phase. Here we will discuss the two best known examples of superfluids: helium-4 and helium-3 (helium is the only element which doesn’t freeze in to a solid, no matter how cold it gets). Although helium-4 and helium-3 are both superfluids, the physics behind their behaviour is different.

The reason why the two superfluids have such different physics is due to the difference between the two most fundamental categories of particles, bosons and fermions. Bosons have integer spin, whereas fermions have half-integer spin. Protons and neutrons, which make up the nuclei of atoms (see diagram), have spin ½. The nucleus of helium-4 has two of each, adding up to integer spin overall—it’s a boson. Whereas the nucleus of helium-3 has two protons but only one neutron, so overall it has half integer spin, making it a fermion instead. Both atoms are helium because they have the same number of protons, two each, and thus are chemically the same (they are called different isotopes of helium).

If you’ve already read our article about spin and the Pauli exclusion principle, you’ll know that in a system consisting of fermions, we can only ever place one fermion at a time into each quantum state of the system. In practice, this means that at low temperatures we will find one fermion in the lowest energy state of the system, one in the next lowest energy state, etc. But bosons do not have to obey this principle, so at low temperatures every boson in a system can be packed into the lowest energy state. This means that as a gas of bosons is cooled down, at some low temperature we will find a significant fraction of the particles are now occupying the zero energy (i.e. not moving) state. This is yet another example of a phase transition. The phase in which this happens is called a Bose-Einstein condensate, named after Albert Einstein and Satyendra Nath Bose, who first theorised its existence in 1925.

Helium-4 atoms are bosons, so clearly liquid helium-4 should become a Bose-Einstein condensate at low enough temperatures. But Bose and Einstein didn’t account for interactions between the atoms, and once we account for these we find that helium-4 will also superflow (flow without viscosity i.e. become a superfluid). This is what happens to helium-4 when it’s cooled to around 2K (-271°C), as discovered in 1938.

The existence of the helium-3 superfluid phase was theorised in the 1960s before it was discovered in 1972. The superfluid behaviour of helium-3 can’t be explained in the same way as the superfluid behaviour of helium-4 because it is a fermion, not a boson. If you’ve read the section above, however, then you already know why helium-3 becomes a superfluid—electrons, which are also fermions, can form Cooper pairs at low temperature and move without electrical resistance. Helium-3 atoms can also form Cooper pairs, which allows them to flow without viscosity. This only happens below an extremely low temperature of 2.7 mK or 0.00027 K.