Category Archives: Quantum

It is undoubtable that quantum mechanics has dramatically changed the world of physics and greatly influenced many other fields as well — including biology, chemistry, and computer science. But besides that, it has expanded outside the realm of science by posing interesting philosophical questions.

The goal of science and philosophy is actually very similar – they are trying to provide answers to fundamental questions. However, they adopt widely different approaches to reach these answers. Whereas philosophy only utilises rational argumentation and critical thinking, science relies on observable evidence and the rigour of the scientific method. To demonstrate that, I am going to use one of the oldest philosophical questions as an example: What is the world made of? Philosophers had been trying to answer this question for centuries, but without much success. Many of them thought they knew the answer – Thales, for instance, argued that everything was made of water. He was wrong — just like many other philosophers of that time — because philosophy is inherently bad at answering such questions.

Today, we know the answer due to modern physics — every object around us is made up of quarks and electrons. And even though this may not be the final answer — string theory proposes even smaller building blocks — there is no doubt that science has made more progress in unravelling the nature of our universe in a few decades than philosophy has in centuries. When it comes to studying nature, the scientific method is by far the best tool we have.

This certainly does not mean that philosophy is unimportant or redundant in today’s world. In fact, the opposite is true. While science is much better at providing correct answers, it cannot make sense of its own discoveries. Thanks to science, we know that humans have evolved in the process of natural selection or that the universe is going to come to an end, but we still need philosophy to decide what to make of these discoveries. Every scientific discovery brings forth a multitude of questions which can only be answered by philosophical discourse, not by conducting more experiments. For instance, biologists have discovered that you could theoretically create your exact copy — or thousands of such copies — by the process of cloning, but it is up to each of us to decide how we feel about that.

But this chapter is not about quarks, electrons, or the relationship of science and philosophy. Instead, I would like to talk about a particular philosophical question, which remains open to this day — the question of free will. Free will is usually defined as the ability to consciously control one’s actions. For instance, let us imagine that you have a day off and decide to go for a trip. Most people would argue that this decision was completely voluntary — you could have just as well decided to stay home. There is no reason to think that you were somehow predetermined to decide this way. We assume to have free will because we naturally feel that way. But is it really the case?

Initially, this issue was purely philosophical. But just like with the question of the fundamental substance, it has eventually expanded into the domain of science, which first contributed to the free will debate with the arrival of Newtonian physics. As you may recall from early chapters, classical physics states that the universe is inherently deterministic. In other words, it presumes that we can, in theory, gather all information about the present state of the universe and use it to predict the future with absolute certainty.

This so-called “clockwork universe” was a crucial factor in the development of the debate, as it makes free will impossible. Your decision to go for a trip was inevitably caused by the specific arrangement of molecules in your brain, which was predetermined at the beginning of the universe just like everything else that has ever happened. Every action has a predictable reaction, and your decisions are mere consequences of a huge chain of actions going back all the way to the Big Bang. Any voluntary decision that is not based on anything that came before it would violate the laws of Newtonian physics, as it would create a new, independent chain of actions and reactions. Such a universe would be both boring and unimaginably frightening. Anybody possessing enough information about the current state of the cosmos would be able to predict all of your future decisions with absolute certainty. All events in the universe would simply play out according to a precisely written script. (Some philosophers, called “compatibilists”, argue that determinism and free will are still compatible. This belief is caused by the fact that they use a different definition of free will than the one people are used to.)

Luckily, quantum mechanics brings liberation from the clockwork universe. How could we predict the future position of a particle when we are not even able to precisely know its current position? In our universe, one can only use probabilities to make assertions about the future. The non-determinism of quantum mechanics is often viewed negatively, as it undermines the simplicity of Newtonian physics. However, only in a probabilistic universe is the future a mystery.

Some physicists even went so far as to use the uncertainty principle as undeniable evidence of free will’s existence. However, that is nothing more than wishful thinking. While it is true that quantum mechanics could perhaps allow for free will to exist — certainly more than Newtonian physics — we know far too little to make any definite conclusions. As of today, we are left with two equally valid options:

The first option is quite straightforward — in spite of the uncertainty principle, our brains are still purely deterministic. One may say that, after all, quantum phenomena only apply to much smaller objects than the brain’s neurones or synapses, so why should they have any significant effect on the way we think? While this could be true, it is certainly not a conclusive argument. Quantum effects are exclusive to the microworld, but that certainly does not mean they have no effect in the macroworld. For instance, in Schrödinger’s thought experiment, the decay of a single nucleus — which is governed purely by probabilities — is scaled up to the macroworld, so that it affects whether the cat dies or lives.

This brings us to the second option, which is a lot more interesting than the first one. Consider for a moment that quantum mechanics really does play a significant role in decision making. What would it mean for the notion of free will? You could assign a certain probability value to all of your future decisions. For example, just before you decided to go for a trip, there was a 60 percent probability that you were going to decide this way, and a 40 percent probability that you were going to stay at home. (Of course, this is a huge oversimplification. Every decision could have hundreds of possible options with constantly varying probability values.) This would make your decisions non-deterministic and therefore completely unpredictable. But is this really free will? Would these decisions be truly “yours”? After all, quantum mechanics is completely random, and is randomness synonymous with freedom?

These questions have no definite answers — at least not yet. We have no idea whether quantum mechanics plays a crucial role in our brains, and if so, nobody knows what to make of it. Right now, it is simply up to each of us to decide what we believe and whether this question even matters to us. However, one thing is certain — thanks to quantum mechanics, the free will debate remains open to this day.

Ever since its establishment, quantum mechanics has faced attacks of many physicist, who could not accept this theory and therefore tried to prove that it is incorrect. Albert Einstein was, without doubt, the most famous of the physicists who tried to disprove quantum mechanics. Paradoxically, he was one of its founders, since he pointed out the wave-particle duality of light. Einstein did not like the unpredictability which was brought to the world of physics by the wave function, and to show his opposition towards the uncertainty of the quantum world, he uttered the famous sentence “God does not play dice with the universe”.

However, quantum mechanics survived all the attempts to refute it. Today, it is one of the best-tested physical theories. Nonetheless, one of the most significant quantum mechanical phenomena, the principle of quantum superposition, is still surrounded with many questions that nobody can answer – as demonstrated by the aforementioned Schrödinger’s paradox (Schrödinger’s cat). Now, however, we are going to discuss a different well-known paradox of quantum mechanics – the EPR paradox.

The EPR paradox is a thought experiment in which three prominent physicists (Albert Einstein, Boris Podolsky and Nathan Rosen) sought to demonstrate the incompleteness of quantum mechanics. Let us say we create a pair of entangled particles and immediately isolate them from their surroundings, so that the wave function of the pair does not collapse. One of the particles is then transported to the Moon, the other one is left here on Earth. Quantum mechanics states that if one observes either of the particles (the one on Earth, say), the wave function of both particles collapses immediately. This means that the particle on the Moon knows straight away when the particle on Earth is observed.

However, the creators of the EPR paradox did not like this “spooky action at a distance” (in Einstein’s own words), since they thought it contradicted Einstein’s theory of relativity. According to special relativity, no information can travel through space faster than light. This rule is fundamental to the theory of relativity, and strange things would begin to happen if it were violated – if information travelled faster than light, to some observers it would seem as if it reached its destination before it had even been sent!

The fact that quantum entanglement seemingly violates this rule made the authors of the EPR experiment think that quantum mechanics was wrong. Instead of the uncertainty of the quantum world, they proposed the so-called hidden variables. Einstein assumed that entangled particles always “agree” in advance on which state each of them takes, which would eliminate the unlovable “spooky action at a distance”. If this hypothesis were true, it would mean that the basic principles of quantum mechanics, like quantum entanglement or quantum superposition, are merely an illusion.

A few years later, a physicist John Stewart Bell came with a relatively complex experiment that could prove whether particles actually communicate faster than light, or whether hidden variables exist, as proposed by Einstein. This experiment includes measuring the spins of entangled particles in various directions by two measuring devices. To the satisfaction of physicist fighting for quantum mechanics, the experiment confirmed that no theories that include hidden variables can replace quantum mechanics. This proved that the authors of the EPR paradox were wrong.

However, if quantum entanglement is real, how could we explain the seeming contradiction with special relativity? The trick is that it is actually impossible to transmit information through entangled pairs, since entanglement is based on probability.

Let us say we have an entangled pair of photons with opposite spins which we want to use to transmit information at a speed that is greater than the speed of light. We agree with the receiver of our message in advance how the message would be encoded – one of the spins could be assigned the value YES (1), the other spin could be assigned the value NO (0). Then, we split the photons – we keep one photon and send the other one to our receiver. In case our photon is observed, the combined function of both photons collapses and the message is automatically sent. Say we want the receiver’s photon to show the value YES, which means that we must influence the spin our own photon (in this case our photon must have the value NO). The problem is, however, that there is no way of determining the spin of our photon, remember that wave function collapse is completely random. This basically means that if we sent the message, the receiver has a 50 percent chance of receiving the value YES and a 50 percent chance of receiving the value NO. Obviously, such communication does not make any sense.

Some phenomena of quantum mechanics might have a huge impact on human technology in the future, particularly in the form of quantum computers. A quantum computer is a computer using quantum superposition and quantum entanglement to improve its computing power. How?

In order to understand quantum computers, we first need to take a look at classical computers. The basic unit of information of classical computers is a bit. A bit can take one of two values: 1/0 (yes/no, on/off). Two bits may take one of four values (11/10/01/00), three bits one of eight values, four bits one of sixteen values, and so forth.

Quantum computers use a slightly different unit of information – a qubit (quantum bit). Qubits are similar to bits, but with one significant difference – due to quantum superposition, a qubit may take more values simultaneously! A qubit can thus be in a superposition of values 1 and 0. We could, for instance, create a qubit using an electron’s spin. Spin ½ could be assigned the value 1, spin −½ the value 0 (or vice versa). As long as the electron is not observed, its qubit has both possible values.

However, if we add another qubit, the whole situation becomes even more interesting. Due to quantum entanglement, both qubits enter a superposition of four states. Qubits now take all four possible values (11, 10, 01 and 00) simultaneously. If we add another qubit, the whole quantum system of these three qubits can take eight values at the same time, and so forth. Each time we add a qubit, the number of possible superposed states doubles. The main difference between a classical and a quantum computer is thus in the number of states it takes. While any set of bits can only take one possible value at a time, the same set of qubits can take all of these values simultaneously.

But what is the consequence of this difference? Speed. Quantum computer is capable of solving certain tasks even a million times faster than a classical computer of comparable size – for instance, a quantum computer composed of just twenty qubits can take 1048576 states simultaneously!

This may sound terrific, but there is a downside. Despite its tremendous speed, quantum computers will probably never entirely replace classical computers. The reason is simple – any time a quantum system is observed, the wave function of this system collapses. This means that anytime we tried to use a quantum computer, there would inevitably be an interaction between us and the computer. This interaction would cause the superposition within the quantum computer to collapse, and its qubits would suddenly become mere classical bits.

Unfortunately, quantum computers are only suitable for specific, usually complex computations. During the computation, they must be isolated from their surroundings to prevent the superposition of their qubits from collapsing. A quantum computer basically divides each problem into many simpler calculations, which it then solves in parallel. Once the computation is finished, the computer is observed, which causes its superposition to collapse, and it provides us with just one result.

The Heisenberg uncertainty principle states that there are certain pairs of variables whose values cannot be known simultaneously. As mentioned earlier, an example of such variables is the pair momentum and position. However, this pair is not the only one which obeys the uncertainty principle. Another such a pair is energy and time:

𝚫𝐄 ⋅ 𝚫𝐭 ≥ ħ/𝟐

Let us say, for instance, that we have a measuring device around which we send a photon. We want to measure the energy of the photon and the time in which the photon has passed the measuring device. However, every particle obeys the uncertainty principle for time and energy, so the more precisely we measure the energy of the photon, the greater uncertainty there is about the time it passed the measuring device.

But what happens if we apply this uncertainty to vacuum? Vacuum is by classical physics defined as empty space (space where there are no particles), therefore, its energy should be zero. However, the uncertainty principle for time and energy states that there is always at least a tiny amount of uncertainty regarding the energy of every system, which means that one can never be sure that the energy of vacuum is truly zero. This means that even vacuum itself can obtain non-zero amount of energy for short periods of time. These deviations in the energy of vacuum are called vacuum fluctuations. The question is: What is this temporary energy caused by vacuum fluctuations used for?

It turns out that it is used to create a peculiar new type of particles – virtual particles. These virtual particles of vacuum fluctuations are created spontaneously everywhere in the universe and usually exist for very short periods of time. Each virtual particle may never be created by itself – it is always created in pair together with its antiparticle. As one might expect, they annihilate after a short period of time.

The equation above shows that the greater the uncertainty in energy, the smaller the uncertainty in time. This means that the more energy a given virtual pair “borrowed”, the sooner the particles of the pair have to annihilate. When a virtual pair annihilates, no energy is created, the law of conservation of energy (energy cannot be created out of nothing) is thus not violated. Virtual particles and antiparticles simply “borrow” energy which they soon return.

Virtual particles might not always have the same properties as their classical counterparts. A virtual electron, for instance, might not have the same mass as a classical electron. Moreover, virtual particles cannot be observed directly. We can, however, observe their impact on the environment around them. Under certain conditions, they can even be transmuted into classical particles, as we shall see in the following chapters.

Casimir Effect

The Casimir effect is a physical phenomenon which proves the existence of virtual particles. It was predicted in 1948 by a Dutch physicist Hendrik Casimir based on the uncertainty principle for time and energy.

Casimir correctly assumed that if we put two parallel uncharged plates just a few nanometres apart, they will attract as a consequence of vacuum fluctuations.

As we have learned in the previous chapter, virtual pairs of particles and antiparticles are being created constantly between and around the plates. However, for a virtual pair to be created between the plates, its wave function must have a relatively small wavelength, since greater wavelengths do not fit between the plates. Consequently, less virtual particles are being created between the plates than in other places around the plates, where particles of arbitrary wavelengths can be created. This results in a greater pressure on the outside of the plates, and the plates start drawing closer.

Hawking Radiation

Everybody is presumably familiar with gravity to some extent. Gravity is an omnipresent attractive force that keeps us on the Earth. But it is also the force that keeps the Earth in orbit around the Sun and the force that makes our whole solar system orbit the centre of the Milky Way (the galaxy which we inhabit).

For a long time, people mistakenly believed that gravity acts merely on particles with mass. Later, however, it was revealed that even particles with zero rest mass (photons, for instance) are influenced gravitationally. Light, which is moving through the universe at the greatest possible speed – according to the special theory of relativity – does not significantly feel gravity in most cases. Nonetheless, there are objects of extreme masses within the universe whose gravitational fields are so incredibly strong that even light cannot escape – black holes.

Generally, the closer an object is to a gravitational field, the greater is the gravity acting on it. Therefore, there has to be an area shaped like a sphere around every black hole beyond which the gravity is so immensely strong that even light does not have any chance of escaping. Scientists call this area the event horizon.

When a famous British physicist Stephen Hawking studied quantum mechanical phenomena near the event horizon in 1974, he came up with a fascinating theory – every black hole should constantly emit electromagnetic radiation. Today, his theory is widely accepted, and this type of radiation is known as Hawking radiation.

Let us imagine a pair of virtual photons which is created near the event horizon in such a way that one of the photons appears directly beyond and the other photon directly in front of the event horizon. The first photon is irrecoverably absorbed by the black hole, while the other photon narrowly escapes this fate. However, since it is a virtual particle, it ought to be destroyed immediately. Nevertheless, virtual particles are destroyed purely by annihilation, which cannot occur, since the escaped photon “lost” the other photon inside of the black hole, which means they cannot collide.

The question therefore is: What happens to the escaped photon? Something seemingly impossible – it becomes a classical photon and leaves the surroundings of the black hole as Hawking radiation.

However, there is a problem. Photons cannot be created out of nothing, the law of conservation of energy must be adhered to. One does not have to worry about the law of conservation of energy as long as the photons are virtual, since the “borrowed” energy of vacuum fluctuations to make these photons is returned after a very short period of time.

But in the case of Hawking radiation, annihilation never occurs – the virtual photon has to obtain energy in order to turn into a classical photon. Where does it get the necessary energy? From the black hole itself. This, however, has surprising consequences. As the black hole gives away its energy to every single virtual photon it emits, its mass decreases – the black hole starts evaporating. The smaller the black hole, the faster it evaporates as a consequence of Hawking radiation.

However, this effect is completely negligible for black holes of cosmic sizes, which usually absorb tremendous amounts of matter, steadily gaining energy instead of losing it.

Every object around us is made up of massive particles. Collectively, we refer to these particles as matter. However, there is a deeply similar entity in the universe that we do not encounter on a daily basis – antimatter. Antimatter is composed of antiparticles, which have the same mass as their particle counterparts but are oppositely charged. For example, the antiparticle of electron, called positron, is positively charged, while the electron is negatively charged. When a particle comes in contact with an antiparticle, both of them are destroyed while releasing an enormous amount of energy. This process is called annihilation.

Let us imagine a situation where a particle collides with its antiparticle, electron with a positron, for instance, while the electron has a spin opposite to the spin of the positron at the time of the collision, so that their overall spin is zero. Once they collide, annihilation occurs instantly. In this case, the annihilation energy is released in the form of two photons of gamma radiation. Let us label the photons as photon A and photon B.

As mentioned earlier, spin represents the intrinsic angular momentum. That is to say that spin obeys the law of conservation of angular momentum, which states that the total angular momentum of a system does not change over time. In other words, if the total spin of the system of the electron and the positron was zero, the total spin of the photons A, B has to be zero as well. Photon A therefore must have a spin that is opposite to the spin of photon B. For illustration, let us label the spins of the photons as spin 1, spin 2.

However, remember that unless a quantum object is observed, it is in a superposition of all possible states. Photon A is therefore in a superposition of spin 1 and spin 2. The same thing applies to photon B. The spin of neither of the photons is defined, but it is given that the spin of one photon must be opposite to the spin of the other photon.

If somebody observes one of the photons (say, photon A) and tries to measure its spin, its wave function collapses, and the photon obtains only one spin (say, spin 1). To fulfil the law of conservation of angular momentum, immediately after the wave function of photon A has collapsed, the wave function of photon B must collapse as well, so that the total spin of photons A and B stays zero.

In other words, the photons are in a state wherein an observation of photon A immediately influences the state of photon B, regardless of the distance between the photons. This state of a kind of superposition, where observation of one object determines the state of another object, is called quantum entanglement. Mathematically we can write the entangled state of photons A, B with spins 1, 2 as follows:

|𝛙⟩ = |𝟏_𝑨⟩|𝟐_𝑩⟩ + |𝟐_𝑨⟩|𝟏_𝑩⟩

Bosons are particles with integer spin (1 ħ, 2 ħ, 3 ħ, etc.). They function as particles that transmit interactions, they are therefore often referred to as force carriers. The most famous boson is indisputably the photon. Bosons also include the W and Z bosons, which are accountable for weak nuclear force (the interaction that causes radioactive decay), gluons, accountable for strong nuclear force (the interaction that holds particles inside of the nucleus of an atom together), and the famous Higgs boson.

Bosons do not obey the Pauli exclusion principle, since they are described by symmetrical wave functions, which means that more bosons can occupy the same quantum state. Bosons basically “crave” to be in the same state as other bosons. This property is responsible for the existence of multiple fascinating phenomena.

Let us start with a laser beam. Laser is a device emitting an immensely narrow beam of light, whose photons have the same frequency and are in phase. This is completely different from classical lightbulbs, which produce light of dozens of frequencies in all directions.

Lasers exploit the fact that photons belong to bosons. Within a laser, there are millions of atoms whose electrons are exited from the ground state to a higher energy level using electric current. Some of these electrons consequently emit energy in the form of photons and jump back to the ground state. The emitted photons then fly around the remaining excited electrons and stimulate the emission of other photons, while all of these photons enter the same quantum state (i.e. have the same frequency and are in phase). Once there are enough emitted photons, they leave the laser in the form of a laser beam.

Another mesmerizing instance of bosons in action can be observed when cooling a group of helium-4 atoms to extremely low temperatures – no more than two degrees above absolute zero. Every helium-4 atom is composed of an even number of fermions. This, however, makes the atom itself a boson, which means that it does not obey the Pauli exclusion principle.

All helium-4 atoms therefore behave just like other bosons – they wish to be in the same quantum state. Unfortunately, they cannot achieve that under normal conditions, as their wave functions look nothing alike. Nevertheless, once they reach temperatures just above absolute zero, their wave functions start spreading and overlapping. Eventually, they enter the same quantum state and the wave functions join into a single unified wave function which describes the entire group as a whole. In other words, quantum behaviour starts to transform into the macroworld!

Such a state of matter, in which atoms enter the same quantum state, is called the Bose-Einstein condensate. In some cases, this condensate behaves unlike any other state of matter. For instance, if one fills a vessel with cooled helium-4 atoms, they gradually creep along the walls of the vessel and escape, seemingly defying gravity.

Fermions are particles with half-integer spin (1/2 ħ, 3/2 ħ, 5/2 ħ, etc.). They serve as the fundamental building blocks of matter. Fermions can be divided into two groups – leptons and quarks. The electron, for instance, is a lepton. The proton and the neutron, however, do not belong to either of the two groups, as they are not elementary particles – both of them are made up of three quarks. Nevertheless, they are still considered fermions. In fact, all composite particles that consist of an odd number of fermions also belong to fermions.

As we have learned in the previous chapter, all fermions have an antisymmetric wave function. This may seem irrelevant, but the opposite is true. Antisymmetric wave functions bring far-reaching consequences in the form of the Pauli exclusion principle.

If we take a look at the structure of an atom, we find out that each atomic orbital is occupied by two electrons at most. This is somewhat peculiar, since everything in the universe has a tendency to stay on the lowest possible energy level. We may notice this when observing the behaviour of object in the gravitational field of the Earth – objects fall down to decrease the value of their potential energy. But electrons seem to just ignore this rule entirely – otherwise they would all gather in the orbital with the lowest energy. What prevents them from doing so?

The Pauli exclusion principle states that no two fermions can be in the same quantum state, which means that each fermion must have at least one property (spin, momentum, etc.) different from all the other fermions. Why? The Pauli exclusion principle is associated with antisymmetric wave functions of fermions.

Let us consider two electrons that are described by a combined wave function ψ_(1,2). Recall that when swapping the electrons, the sign of the wave function is changed due to its antisymmetric nature: ψ_(1,2) = −ψ_(2,1). But also recall that any particle can be in all possible states at once due to the superposition principle, which means that if the given electrons can be described by the wave function ψ_(1,2) as well as the wave function ψ_(2,1), they are in a superposition of both of these wave functions. This superposition looks as follows:

𝛙 = 𝛙_(𝟏,𝟐) − 𝛙_(𝟐,𝟏)

However, we can see that if the two wave functions were the same (i.e. if the electrons were in the same quantum state), the equation above would be equal to zero – the electrons basically would not exist! This is not possible, of course. And the Pauli exclusion principle prevents that – it simply ensures that the equation ψ = ψ_(1,2) − ψ_(2,1) is never equal to zero, since no two fermions will ever be in the same state.

But what does it have to do with electrons inside of an atom? Electrons belong to fermions, the Pauli exclusion principle therefore applies to them. If all electrons gathered in the orbital with the lowest energy, they would violate this crucial principle, as they would all be in the same quantum state. But there is one more crucial fact to be explained – why are there at most two electrons in each orbital and not just one?

This phenomenon can be explained using spin. The spin of an electron can take two different values: ½ or −½. When electrons are in the same orbital, they have the same amount of energy, but they still have different spins – one of them has a spin of ½, the other one has a spin of −½. This way, they do not violate the Pauli exclusion principle, as different spins mean different quantum states. However, no other electron can be found in this orbital, because it would inevitably be in the same quantum state with one of the original electrons.

The existence of the Pauli exclusion principle is crucial for stable structures to form. If it did not exist, the universe would be a widely different place. Molecules, for instance, would not form, since atoms would simply not be able to bind to each other.

When describing objects from the macroworld, we often use words like “identical” or “same”. We could proclaim, for instance, that two mobile phones of the same model are the same. The problem is, however, that no two objects from the macroworld are actually “the same”. There is at least a slight difference between any two objects from the macroworld. With the mobile phones mentioned earlier, the difference is not visible at first sight, since it is on a molecular level. In addition, one can always simply differentiate between the phones by marking them (for example, one can paint one of the phones blue and the other one red).

In the microworld, however, the words “identical” or “indistinguishable” have a completely different meaning. Any two electrons (protons, photons, etc.) are absolutely identical and there is no way of telling them apart. One cannot mark them in order to make them different either (it is simply not possible to “paint” an electron, since colour has no meaning in the microworld). It therefore makes no sense to refer to two electrons as “the first electron” and “the second electron”, since there is no way of telling which one is which.

Let us consider two identical particles, one of them is described by the wave function ψ₍₁₎, the other is described by the wave function ψ₍₂₎:

We could describe these two particles by a combined wave function that is a combination of the two original wave functions ψ₍₁₎ and ψ₍₂₎. This wave function would take a form ψ_(1,2) = ψ₍₁₎ ψ₍₂₎:

But what happens if we swap the particles (i.e. the particle that was originally assigned the wave function ψ₍₁₎ is now assigned the wave function ψ₍₂₎ and vice versa) and describe them by a combined wave function in the form of ψ_(2,1) = ψ₍₁₎ψ₍₂₎? The particles are indistinguishable, so we should not be able to spot any difference after we swap the particles and the system should look exactly the same as before the particles were swapped. One can achieve that only if the wave function ψ_(1,2), which describes the system before the particles are swapped, is identical to the wave function ψ_(2,1), which describes the system after we swap the particles, therefore:

𝛙_(𝟏,𝟐) = ±𝛙_(𝟐,𝟏)

In some cases, however, it may happen that the wave function changes its sign after the particles are swapped. In case this happens, the wave function is considered to be antisymmetric. If the sign remains preserved, it is a symmetric wave function. Bosons are described by symmetric wave functions, antisymmetric wave functions are typical for fermions.

Have you ever wondered how magnets work? How is it possible that some materials, like iron, show magnetic properties, while other materials, like wood, seem to ignore magnetism entirely? It turned out that the answer to these questions lies in a strange property of all particles called spin.

One could say that there are two types of momentum in the macroworld – “classical” momentum, which objects acquire by moving in a certain direction, and angular momentum, better known as rotation. However, objects from the microworld have an additional type of momentum – intrinsic angular momentum or spin. Spin is often compared to classical rotation (hence the name “spin”). However, this comparison is not accurate, since objects with spin do not actually rotate, the rotation is purely “intrinsic”.

Spin is typical for elementary particles, composite particles and atomic nuclei. The unit of spin is the reduced Planck constant (ħ). Particles with half-integer spin (1/2 ħ, 3/2 ħ, 5/2 ħ, etc.) are called fermions. Particles with integer spin (1 ħ, 2 ħ, 3 ħ, etc.) are called bosons. We are going to learn more about these particles in the following chapters.

But what is the connection between spin and magnetism? It turns out that particles with spin behave like peculiar tiny magnets by generating weak magnetic fields. That is why objects from the macroworld, which are composed of many of such “small magnets”, are magnetic. But that does not explain why only a handful of materials are magnetic, when all macroscopic materials are made up of these tiny magnets. How is that possible?

The reason is that magnetic fields generated by individual particles (mostly electrons, whose magnetic fields are much stronger than those of protons or neutrons) often cancel out, which in turn makes most materials non-magnetic.

For instance, if an atomic orbital is completely filled, electrons in this orbital have opposite spins, which causes their magnetic fields to cancel out. This means that no atom with filled or almost filled orbitals can be magnetic. For an atom to be magnetic, it must have half-filled orbitals so that the magnetic fields of individual electrons reinforce one another.

However, not all materials made up of magnetic atoms exhibit magnetic properties. This is due to the configuration of individual atoms. Many materials have their atoms arranged so that their magnetic fields cancel out. Only a fraction of materials have the atoms arranged so that their magnetic fields mutually reinforce. This is why magnetic materials are so rare.

In the previous paragraphs, particles were compared to tiny magnets. However, this comparison is not completely accurate, because magnetic fields created by particles behave rather oddly. We can demonstrate this on a simple experiment. Say we have two axes: x and y, which are perpendicular to each other. Now let us take a particle which has a spin of 1 pointing in the direction of the x-axis (i.e. if we were to measure its spin in the direction of the x-axis, we would get 1). But what happens if we try to measure its spin (magnetic field) in the y-axis? If we took a classical magnet, pointed it in the direction x and conducted the same experiment, we would measure no magnetic field pointing in the y direction, of course (since the x-axis is perpendicular to the y-axis). However, particles behave in a completely different way. If we measure the spin of a particle in the y direction, half the time we get spin 1, the other half of the time we get spin −1. Even if we try to measure the spin in different directions, we always get either 1 or −1. However, the average of the values we get is always equal to the value we would expect to get with classical magnets. We can demonstrate this rule on our particle. The average of the values we got (half the time 1, half the time −1) is equal to zero, which is the value we would get with a normal magnet.

About 150 million kilometres away from us, there is a huge sphere of hot plasma, which we call the Sun. Just like any other star, the Sun makes its energy by colliding lighter atomic nuclei to form a heavier element. This process, called nuclear fusion, is crucial for the existence of every single star in the universe.

However, there is a problem. The colliding nuclei are all positively charged, which means that they repel each other electrically. How do the nuclei fuse, then? There is another force – the strong nuclear force – which brings them together, but only when they are really close to each other to begin with. Therefore, the nuclei must have a huge energy (and thus velocity) in order to approach each other to the point where the attractive nuclear strong force surpasses the repulsive electrical force for nuclear fusion to occur. But when the temperature of the Sun was ascertained by its spectrum, it came to light that it does not even remotely reach the values necessary for nuclear fusion. In other words, the Sun simply should not shine whatsoever. This conclusion is obviously wrong – the Sun evidently shines, for which we owe to a peculiar phenomenon of quantum physics – quantum tunnelling.

Quantum tunnelling is a phenomenon wherein particles or even whole atoms have a certain probability of surpassing a barrier, even though they do not have enough energy to surpass it, which is unambiguously against the principles of classical physics. This phenomenon may not seem that peculiar at first sight, but the opposite is true. It would probably be quite strange if a person who run up against a wall appeared on the other side of the wall or even inside the wall. However incredible it may sound, this is essentially what happens to objects from the microworld during quantum tunnelling.

Quantum tunnelling can be explained using the principle of quantum superposition and the uncertainty principle. How? According to classical physics, the Sun does not have the sufficient temperature for atomic nuclei to approach each other enough for fusion to occur. However, the principle of quantum superposition states that the nuclei can be in more places at once (due to their wave nature), so there is a certain probability of them approaching enough and fusing. According to the Heisenberg uncertainty principle, on the other hand, there is always some uncertainty regarding the momentum of an object, so from time to time, one or both nuclei obtain an immense velocity (momentum) and fuse.

Quantum tunnelling is one of a few phenomena of quantum mechanics whose consequences we can hugely feel in the macroworld as well. The structure of our own bodies, for instance, is determined by the DNA molecule. However, it has been theorised that protons within this molecule can experience quantum tunnelling and therefore change our genetic makeup! These random genetic mutations caused by quantum tunnelling may even be linked to the existence of cancer, but more research is needed. Tunnelling also occurs during radioactive decay or in flash discs.