Category Archives: Quantum

Let us say we are conducting an experiment where we are sending electrons through a narrow slit. In a certain distance behind the slit, there is a detection plate, which detects the position of individual electrons that strike it. We already know from the previous chapters that we cannot predict where any individual electron ends up on the plate (because of superposition). We can, however, know the probability of an electron ending up in a certain place on the plate if we know its wave function.

If we make the slit smaller, we probably intuitively expect the electrons to fall into a narrower section on the plate. Let us say we start with a relatively wide opening which we taper gradually. At first, our prediction is correct and the electrons indeed start falling into an increasingly narrower section. At some point, however, the opposite begins to happen. If one continues to make the slit smaller to the point where it is considerably narrow, the electrons start spreading again.

This phenomenon is a consequence of the so-called Heisenberg uncertainty principle, which was introduced by Werner Heisenberg in 1927. The uncertainty principle states that there are pairs of physical properties whose precise values cannot be known simultaneously. The more precisely we know one property, the more uncertainty there is about the other property. The most famous pair of such properties is momentum and position. The uncertainty in the momentum of a given particle multiplied by the uncertainty in the position of this particle is always equal or greater than the value of the reduced Planck constant divided by two:

𝚫𝐱 ⋅ 𝚫𝐩 ≥ ħ / 𝟐
(ħ = 𝐡 / 𝟐𝛑 )

The more accurately one knows the position of a particle, the less information one has about its momentum. Let us go back to the electrons going through a slit. If we make the slit narrower, the uncertainty about the position of the electrons is decreased. Consequently, the uncertainty about their momentum has to be increased. The electrons now have a greater probability of changing their direction (i.e. are deflected sideways) or velocity, leading to them being more spread on the plate.

The Heisenberg uncertainty principle is a mere consequence of the wave function. Let us consider, for example, that we want to measure the momentum of a certain particle as accurately as possible. De Broglie’s equation (λ = h/p) shows that the momentum of a particle depends on the wavelength of its wave function (p = h/λ). Therefore, if we want to ascertain the wavelength, the wave function cannot be too localized, since the wavelength of a localized wave is not precisely determined. On the other hand, if we want to measure the position of a particle, we need a wave that is as localized as possible. Of course, a wave cannot be both localized and spread simultaneously, which means that when measuring the position and the momentum of a particle at the same time, one has to find a compromise in the form of a wave function that is partially localized and partially spread and as such provides relatively precise values for both position and momentum. Such a wave function is called a wave packet.

The uncertainty principle is often mistakenly interchanged with the so-called observer effect, which is a phenomenon that occurs every time a physical system is observed. The observer effect states that any time a system is observed, its state inevitably changes. For example, when ascertaining the position of an object using our vision, photons have to bounce off the object into our eyes, so its position is not the same as it had been before the observation occurred. This phenomenon, however, has nothing to do with the uncertainty principle, since the uncertainty in the position and the momentum of a quantum object exists all the time, regardless of the presence of an observer. We can basically say that even the object itself does not “know” its own position and momentum simultaneously. Therefore, explaining the uncertainty principle using the observer effect is wrong.

Soon after de Broglie introduced his hypothesis to the world, a period which is often referred to as the old quantum mechanics came to an end (1900 – 1925). The basic phenomena of the old quantum mechanics are the quantization of energy and the wave-particle duality. Since 1925 we are dealing with the modern quantum mechanics.

Austrian physicist Erwin Schrödinger in 1925 adjusts de Broglie’s inaccurate theory and assigns a so-called wave function to every quantum object. Temporal and spatial evolution of a wave function is described by a complex equation, the so-called Schrödinger equation. A wave function is denoted by the capital or lower-case Greek letter psi:

𝜳, 𝝍

The wave function is a complex mathematical function in which all the properties of a given quantum objects (momentum, position, etc.) are stored (this is different from the de Broglie’s matter wave, since de Broglie did not assign this property to his wave, moreover, he perceived the wave as a physical object, while Schrödinger’s wave function is merely abstract). This set of properties of a quantum object is called a quantum state. Quantum state is denoted as follows:

|𝝍⟩

The wave function is presumably the most significant idea of quantum mechanics, since most of the phenomena of the modern quantum mechanics are derived from it. Some of these phenomena, especially the principle of quantum superposition, are so different from the ones which are usual for us in the macroworld that it is often very difficult to believe, let alone understand them.

Quantum Superposition

Already when clarifying the phenomena of the old quantum mechanics, it became clear that the applicability of a certain phenomenon to the macroworld does not necessarily mean the applicability of this phenomenon to the microworld. One of the basic rules of the macroworld is that each object has only one position and one velocity. In other words, it is simply impossible to travel from Germany to the UK at 60 miles per hour while flying from Europe to Australia at eight times that speed. Astonishingly, this rule does not seem to apply to the quantum world, and objects from the microworld may therefore be in many places at once and do many things at once!

When conducting the double-slit experiment, an interference pattern is created only when a wave (wave function) from one slit interferes with a wave that passed through the other slit, as I have already described in the previous chapters. If we were to send waves (whether electromagnetic waves or matter waves) only through one slit, an interference pattern would not be created, of course. Let us imagine a situation where we are conducting the double-slit experiment with massive particles, such as protons, but with one small variation – we are sending only one proton at a time, so the wave functions of individual protons cannot interfere with each other. As strange as it may sound, even in this case, there is an interference pattern.

The entire classical physics is based on the idea of the so-called determinism. The basic principle of determinism is that the future is predictable and that the only thing necessary to predict the future evolution of the universe is having enough information about the present. For example, we can predict the next solar eclipse by having enough information about the motion of the Moon. The entire deterministic physics is based on this condition. Another idea of determinism is that identical conditions lead to identical results. For instance, if we were to shoot two identical bullets from a gun under the same conditions (i.e. in the same direction, at the same temperature, etc.), both bullets would hit the same place. However, the quantum world behaves completely differently. If we were to shoot electrons instead of bullets (from a hypothetical electron gun), each of these electrons could hit a different place and each of them could have a dissimilar velocity, even though the initial conditions were identical.

The strange behaviour of the protons from the second paragraph and the unpredictability of the electrons from the third paragraph are both a consequence of a crucial phenomenon of quantum mechanics – the principle of quantum superposition. Quantum superposition states that an object that is not being observed exists in all possible states at once – it is in a superposition. This superposition is a combination of all the states the object could theoretically be in. This means that a particle which is not observed can have multiple velocities and be at multiple places at once.

This may sound rather strange, but if we take the wave function into consideration, superposition starts to make sense. Consider, for instance, the position of an object. The wave function can be imagined as an abstract mathematical wave surrounding a given object. As mentioned before, the wave function contains all the properties of an object, the position of the wave function thereby determines the position of the object itself. This, however, poses a problem. Recall that a wave is not localized in space but instead tends to spread. This property applies to our wave function as well. It follows that as long as the wave function of an object exists, the position of this object is not fully defined and the object is basically located everywhere where its wave function is located. We say that this object is in multiple eigenstates. For a quantum object to have clearly specified position, the wave function must “disappear”. How to achieve that? Simply by observation.

When a quantum object is observed, the so-called wave function collapse occurs. Wave function collapse is the reduction of the wave function to one eigenstate (one position, one velocity). Wave function collapse ensures that one can never observe an object with multiple velocities or positions, since the superposition is destroyed by mere observation. The act of observation thereby does not only identify the properties of a quantum object but also determines them! That basically means that we determine the future of an object purely by observing it (i.e. measuring its properties).

However, there is one important question: How does a quantum object select an eigenstate in which it is located when it is observed? This process is based on probability. The likelihood of a quantum object ending up in a certain eigenstate is determined by its wave function. The wave function is therefore also referred to as the probability wave. From each wave function, a complex number can be extracted, the so-called probability amplitude, which is used to determine this probability. The probability of a quantum object ending up in a certain eigenstate is determined as the square of the absolute value of the probability amplitude. If the probability of a certain process occurring is 50 percent, for instance, the probability amplitude of this process has a value of 𝟏/√𝟐.

Let us consider a situation where we want to ascertain the velocity of a certain electron. Let us say that this electron is in a quantum state that is a superposition of two eigenstates. The first eigenstate assigns the electron velocity 1, the second eigenstate assigns it velocity 2. This superposition of two velocities can mathematically be written as follows:

|𝛙⟩ = |𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝟏⟩ + |𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝟐⟩

As long as the electron is not observed, it has both velocities. The wave function assigns the electron a probability of ending in each of the eigenstates in case of observation. For illustration, let us assign the electron a 75 percent chance of ending up in the first state with velocity 1 and a 25 percent chance of ending up in the second state with velocity 2. Mathematically we can write this using probability amplitudes:

|𝛙⟩ = (3 / 4)^-½ × |𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝟏⟩ + (1 / 4)^-½ × |𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝟐⟩

If we now try to measure the velocity of the electron, its wave function collapses, and the electron obtains just one velocity. Let us say that in the first measurement, the electron has velocity 1. If we repeat the measurement multiple times using different electrons with the same wave function, we randomly get either velocity 1 or velocity 2. In 75 percent of the cases, the electron has velocity 1, in the remaining 25 percent of the cases, the electron has velocity 2. There is no way of knowing for sure which velocity the electron takes in the next measurement.

A quantum object can be in a superposition of an arbitrary number of eigenstates, and each of these states is assigned a certain probability value. The sum of probability values of all eigenstates of a quantum object in a superposition is equal to one. The probability of finding an object in one of its eigenstates is therefore always equal to 100 percent (in other words, if the object exists, it is always going to be somewhere – even though we may not be able to predict where). Mathematical notation (c1, c2, c3 are probability amplitudes):

|𝛙⟩ = 𝒄𝟏|𝐬𝐭𝐚𝐭𝐞 𝟏⟩ + 𝒄𝟐|𝐬𝐭𝐚𝐭𝐞 𝟐⟩ + 𝒄𝟑|𝐬𝐭𝐚𝐭𝐞 𝟑⟩ + ⋯
|𝒄𝟏|𝟐 + |𝒄𝟐|𝟐 + |𝒄𝟑|𝟐 + ⋯ = 𝟏

Let us now go back to the second and third paragraph of this chapter to understand what was happening. The proton in Young’s experiment is in a superposition, so it actually goes through both slits at once and interferes with itself! If we put a detector in front of the slits and observe which slit the proton goes through, its superposition is destroyed and the interference pattern disappears. The electron fired from a gun (the third paragraph) is in more eigenstates at once, and therefore has multiple velocities and is in multiple places at once. Only after the impact, when the wave-function collapse occurs, does the electron obtain just one position, which, however, does not have to be identical to positions of other fired electrons.

We do not encounter superposition on a daily basis, since objects from the macroworld continuously interact with their environment which acts as an observer, and therefore wave function collapse occurs constantly.

Quantum superposition is an elementary principle of quantum mechanics. It breaks the deterministic perception of the world. In quantum mechanics, future is determined only within probabilities, and the same conditions often lead to utterly different results.

One might think that probability is present in the macroworld as well. However, the opposite is true. Any seemingly random phenomenon from the macroworld, throwing a dice, for instance, is completely deterministic and any “randomness” is caused purely by our insufficient knowledge of the system. In the case of throwing a dice onto a surface, it is the height of the dice above the surface, the speed of the rotation of the dice, the mass of the dice, the surface roughness of the table, and so on. If we had a powerful supercomputer that would be able to take all these factors into consideration, we would know exactly which value the dice would show at any given time.

This, however, does not hold true in the microworld. In quantum mechanics, instead of the question: “Where is a particle located?” we ask the question: “What is the probability of finding a particle in a certain place?”

Schrödinger’s Cat and Interpretations of Quantum Mechanics

There is no doubt that quantum mechanics is a revolutionary and an exceptionally strange theory. After the establishment of the modern quantum mechanics, physicist divided themselves into several opinion groups. Each of these groups tried to explain the weirdness of the quantum world in a different way, which led to the creation of many interpretations of quantum mechanics. The most acknowledged of these interpretations is the so-called Copenhagen interpretation (this interpretation was used in the previous chapter). Another very interesting interpretation is the so-called many-worlds interpretation. We can demonstrate the difference between these two interpretations on a simple thought experiment.

Let us say we have a box in which an atom of a radioactive element is located. A radioactive element is an element that undergoes decay to lighter elements in a certain period of time. The problem is that one can never know when the decay occurs, since each radioactive atom is described by a wave function that determines only the probability of the atom decaying over time. The probability of the decay occurring increases with time. Thus, the so-called half-life was defined. Half-life is the amount of time after which the probability of an atom decaying is exactly 50 percent. Each radioactive element has a different half-life (ranging from fractions of a second to millions of years). For instance, if we had 100 atoms of an element with a half-life of one year, 50 atoms would have decayed after one year.

Let us go back to our atom in a box. For simplicity, suppose that the half-life of our radioactive element is one day, i.e. if we leave the atom in the box for one day, there is a 50 percent probability of it decaying. However, recall that unless a quantum object is observed, it is in a superposition of all possible states. Therefore, the atom is both decayed and not decayed. In other words, our atom isolated inside of the box is in a superposition of two states – decayed / non-decayed. Only when we open the box and observe the atom, does the wave function collapse occur and the atom “decides” whether it is decayed or not decayed based on the probability given by its wave function (after one day, this probability is 50 percent for both decayed and non-decayed state). Now let us consider a situation where we put a vessel full of poisonous gas and a living cat in the box along with the atom. The whole system is set up so that if the decay occurs, the poisonous gas is released and the cat dies. If the atom does not decay, the gas is not released and the cat stays alive.

If this thought experiment seems familiar to you, it is because we are dealing with the most famous “paradox” of quantum mechanics. The author of this thought experiment is a famous physicist Erwin Schrödinger, the experiment is therefore often referred to as Schrödinger’s cat. With this experiment, Schrödinger wanted to demonstrate the vagueness of the Copenhagen interpretation. It bothered him that the Copenhagen interpretation does not clearly define what it means to “observe” a quantum object. According to him, the Copenhagen interpretation basically says that if an atom is in the superposition decayed/non-decayed, the poison is in the superposition released/not released, which implies that the cat is in the superposition alive/dead until the box is opened. The cat obviously cannot be dead and alive simultaneously. That is why Schrödinger considered the Copenhagen interpretation silly.

However, the authors of the Copenhagen interpretation themselves never saw Schrödinger’s cat as a problem, since they reckoned that the fate of the cat is decided long before the box is opened, since atoms in the air around the radioactive atom “observe” (bump into) it and thereby prevent superposition of the cat. Even the cat herself can observe whether the poisonous gas is released or not, therefore preventing superposition.

Each interpretation explains Schrödinger’s cat a little differently. For instance, the aforementioned many-worlds interpretation assumes that every time two quantum systems interact, the reality is split into multiple parallel “worlds”. The interaction leads to different results in each of these worlds. In other words, everything that can happen does happen in at least one of the worlds. This means that when the box is opened, the whole universe splits into two universes, one of them containing a living cat, the other one containing a dead one!

For centuries, physicists all over the world were leading a heated debate about the nature of light and for many years there were two opinion groups among them. Supporters of the first group believed that light was a wave, while members of the other group believed that electromagnetic radiation had a particle nature. However, quantum mechanics showed that neither of the groups was completely right and that the real answer to this question is much stranger and much more complicated than any of the contemporary thinkers could have ever imagined.

Young’s Experiment

Young’s experiment, often referred to as the double-slit experiment, is a relatively simple experiment. It was used at the beginning of the 19th century to prove that light exhibits wave properties. This experiment exploits two specific properties of waves:

1. If a wave reaches a small opening, it bends. This phenomenon is called diffraction. The size of the opening has to be comparable to the wave’s wavelength for diffraction to occur.
2. When two waves encounter, they do not collide but strengthen or weaken each other depending on what the displacement (“height”) of both waves is. This phenomenon is called interference. For example, when two waves with opposite displacements meet (i.e. a crest of one wave meets a trough of another wave), they cancel out. If interfering waves weaken each other, we are talking about destructive interference. The opposite of destructive interference is constructive interference (waves strengthen each other).

In the double-slit experiment, two slits, which are very close to each other, are used. Light passes through both slits and spreads to the medium behind the opening thanks to diffraction. Due to small distance between the slits, the waves from the first slit meet the waves from the second slit and interference occurs. If we situate a plate detecting the position of individual beams of light that strike it, a specific pattern is created, the so-called interference pattern, which consists of light and dark stripes. Light stripes on the plate are located in places where constructive interference of light waves occurs (waves strengthen one another, thereby increasing the intensity of light incident on these places), dark stripes are caused by destructive interference (waves weaken one another, thereby decreasing the light intensity). If light did not exhibit wave properties, interference pattern would not be created.

Young’s experiment is a simple experiment demonstrating the wave nature of electromagnetic radiation. The original version of this experiment is not related to quantum mechanics, but using its modifications, we can easily prove some of the strange phenomena of the microworld, as we shall see in the following chapters.

Photoelectric Effect

According to the Bohr model of the atom, it is necessary to provide electrons with energy in the form of electromagnetic radiation in order for them to get to orbitals with higher energy. However, if an electron absorbs a wave of high frequency (and thus energy), sometimes this energy is sufficient for the electron to abandon the atom entirely. This phenomenon, when electrons are released from the shell of an atom, is called the photoelectric effect. Electrons that are released in this manner are called photoelectrons.

Let us imagine an experiment where one shines light on electrons inside of an atom, whereby some of these electrons are released from the atomic shell and become photoelectrons. Classical physics states that the energy of photoelectrons should be dependent on the intensity of light, since it assumes that the higher the intensity of radiation (i.e. the brighter the light is), the higher the energy of the electromagnetic wave that is then absorbed by the electrons. Nevertheless, this dependency has not been observed. It was experimentally proven that the energy of the emitted electrons depends purely on the frequency of radiation. Also, the existence of the so-called threshold frequency has been observed. If one provides an atom with light that has lower frequency than the threshold frequency, no electrons are released, again regardless of the intensity of radiation. Classical physics is not able to explain this phenomenon.

Young’s experiment presents a very convincing evidence that light is a wave. However, in order to explain the photoelectric effect, we need to perceive light as a set of particles. Electrons do not absorb electromagnetic waves as predicted by classical mechanics. They absorb particles of light, called photons. Photons are identical to the energy quanta Planck proposed to solve the problem with black-body radiation. Einstein, however, was the first one to realize the particle nature of these quanta, and it was him who managed to clarify the photoelectric effect.

If we perceive light as a stream of particles, the photoelectric effect can be explained quite easily. Increase in intensity of radiation raises the number of photons (quanta) in an electromagnetic wave, but individual photons still carry the same amount of energy. That is to say, if one uses more intense light, the energy of photoelectrons stays unchanged, since electrons may absorb only one photon, in accordance with the third Bohr’s rule. However, by increasing the intensity of light, the number of emitted electrons (photoelectrons) is increased as there are now more photons in electromagnetic waves to be absorbed.

If we wanted to raise the energy of photoelectrons, we would have to raise the energy of individual photons. We could achieve that by raising the frequency of radiation, which is obvious from Planck’s equation E = h ⋅ f (E is the energy of a photon). Quantum mechanics is also able to explain the threshold frequency. Individual photons of low-frequency radiation simply do not have enough energy to release an electron. Therefore, the photoelectric effect does not occur.

Albert Einstein also derived the equation for calculating the momentum of a photon (λ is the wavelength of the electromagnetic wave in which the photon is located):

𝐩 = 𝐡 / 𝛌

When an electron absorbs a photon, it obtains all of its energy. Part of this energy is then used to extricate the electron from the atom (the electron must accomplish work W), the rest of the energy is converted into the kinetic energy of the electron. The energy of a photoelectron can thus be calculated using the following equation:

𝐄 = 𝐖 + ½ 𝐦𝐯^𝟐

While Young’s experiment convincingly demonstrates the wave nature of light, the photoelectric effect sees light as a stream of particles. Therefore, electromagnetic radiation is dual in nature. It has both wave and particle nature.

Matter Wave

After a partial clarification of the strange properties of light in the form of the wave-particle duality comes the year 1924 and a young French physicist Louis de Broglie with a very daring hypothesis. According to this hypothesis, duality does not apply only to light but to every single object in the universe. In other words, de Broglie presumed that all objects, including the ones with mass, are surrounded by a kind of wave, similarly to photons. It is not a great surprise that this revolutionary hypothesis was initially not received very well. Opponents of the hypothesis argued that, after all, matter behaves nothing like a wave.

Eventually, however, it turned out that de Broglie had been right and that the so-called matter wave indeed exists. The presence of this wave can be demonstrated with the help of Young’s experiment, for instance. When conducting the double-slit experiment with massive particles (electrons, for example), an interference pattern is created, which confirms de Broglie’s hypothesis. The relation between the momentum of an object and the wavelength of its matter wave is expressed by the following equation:

𝛌 = 𝐡 / 𝐩

The equation above shows that the wavelength of an object’s matter wave decreases when the momentum of the object increases. In other words, the more massive the object, the smaller the wavelength of its matter wave. This is why objects from the macroworld do not exhibit wave-like properties. Matter waves of large objects have very small wavelengths, which means that if we wanted to prove the wave nature of a large object using Young’s experiment, for example, we would encounter a problem. For diffraction and interference to occur, the size of the slits and the distance between them would have to be significantly smaller than the size of the object itself.

In quantum mechanics, light and matter are dual. Sometimes their wave nature comes to light, other times they show their particle nature. This ground-breaking idea is fundamental to quantum mechanics.

Introduction

Motion. At first sight it is something incredibly uninteresting and trivial. People have been studying motion for thousands of years, but it was not until 1687, when Isaac Newton formulated his three laws of motion, that people finally started to understand it more deeply. Newton’s laws of motion were so ahead of their time that some scientists still consider Newton the most revolutionary physicist of all time. But even Newton’s laws are not perfect, and in 1905 came the special theory of relativity, which brilliantly describes the motion of objects moving at high speeds, formulated by Albert Einstein. But there is another theory that started to develop at the same time. A theory that completely changed our perception of reality. In 1900, the cornerstone of quantum mechanics was laid.

Quantum mechanics deals with objects from the so-called microworld, like particles or atoms. These objects behave nothing like objects of “classical” proportions from the so-called macroworld we ordinarily deal with, and thus cannot be described by classical physics.

In this article, you will be able to explore the world of this ground-breaking theory. And if you at any point struggle to comprehend some its peculiar phenomena, do not worry, you are not the only one. Richard Feynman, one of the greatest contributors to quantum mechanics, once said:

“I think I can safely say that nobody understands quantum mechanics.”

Quantization of Energy

“There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.”

This sentence was pronounced by a famous Scottish physicist William Thomson on the verge of the 20th century, and many contemporary physicists undoubtedly agreed with him. Classical physical theories had been tested many times and seemed to describe reality tolerably. Not until later, when these theories started to fall apart, did come to light how horribly wrong Thomson was. The first phenomenon which classical physics failed to explain is called the black-body radiation.

To understand this phenomenon, it is necessary to know that all tangible bodies in the universe emit energy in the form of electromagnetic radiation (light). The amount of energy emitted by a body depends on several factors, such as temperature or colour of the body. The higher the temperature of a body, the higher the average frequency (and thereby energy) of the light it emits. The reason we usually cannot observe this radiation is that bodies at room temperature emit predominantly light from the infrared spectrum, which is not visible to the naked eye. Visible light is emitted by metals during melting, for example, when their temperature reaches several hundreds of degrees Celsius, making it possible for us to see them glow.

Physicists of the 19^th century were trying to ascertain the spectral composition emitted by a body in relation to its temperature. To accomplish that, they used a simplified model of a body – the black body. A black body is a hypothetical body that has to meet the following two conditions:

1. A black body absorbs all the electromagnetic radiation that strikes it (other bodies absorb merely a certain part of the whole spectrum and reflects the remaining light).
2. A black body stays in thermal equilibrium with its surroundings (i.e. has the same temperature as all the bodies located within the same system).

These conditions ensure that spectrum emitted by a black body is determined purely by the temperature of the body. However, when physicists tried to establish the composition of such a spectrum using classical physics, they obtained a result that did not coincide with reality whatsoever. According to classical physics, a black body would emit the same amount of light of each frequency. However, the higher the light’s frequency, the more energy the light has. A black body would therefore emit huge quantities of energy in the form of high-frequency radiation – infinite, in fact.

This, however, has dire consequences – classical physics thereby basically states that every single object in the universe should immediately emit all of its energy in the form of light from the ultraviolet spectrum. Luckily, the universe does not work that way, otherwise we would not exist.

This realisation was a huge milestone for the evolution of modern science. Physicists were at last unwillingly forced to admit that classical theories were simply wrong. Today, we have an apt name for this huge failure of classical physics – the ultraviolet catastrophe.

The black-body radiation problem was solved by a German physicist Max Planck. He came up with an idea that bodies do not emit electromagnetic radiation continuously, but via small packets called quanta. The size of these quanta is given by the following Planck’s equation:

𝐄 = 𝐡 ⋅ 𝐟
(𝐡 = 𝟔,𝟔𝟐𝟔 ⋅ 𝟏𝟎^−𝟑𝟒 𝐉𝐬)

Electromagnetic wave can essentially be thought of as a set of very small energy “packets” (quanta) whose total energy determines the energy of the wave itself. The size of a quantum is specific for each frequency. From the equation above, it is apparent that radiation of higher frequencies is composed of larger quanta than radiation of lower frequencies. This solves the problem with black-body radiation – it is increasingly difficult for a black body to emit radiation of higher frequencies, as it often cannot “feed” high-frequency quanta with enough energy, and thus sticks with low-energy light.

Quantization of energy is just the very beginning of a whole new world of physics. It presents a fundamental rule to quantum mechanics – as we will learn in the following chapters.

Bohr Model of the Atom

Imagine that you have an object that you then start dividing into smaller and smaller parts. Would you be able to divide the object forever, or would you eventually reach some peculiar indivisible building blocks? Scholars of ancient Greece have asked themselves the same question and eventually have come with a correct assumption: all matter in the universe is made up of very small “grains”. They called these grains atoms (atomos = indivisible).

Later, when scientists thought to have discovered these indivisible blocks, they adopted the Greek name. It was then revealed that atoms are actually not indivisible, but consist of positively charged protons, negatively charged electrons, and neutrons, which are uncharged. However, there was uncertainty regarding the structure of the atom, and the physicist living at the beginning of the 20th century were trying to clarify it.

In 1911, Ernest Rutherford proposed the so-called planetary model of the atom. According to this model, every atom consists of a positively charged nucleus around which orbit negatively charged electrons like planets around stars. However, this model has one major flaw – if atoms obeyed the model, they would be extremely unstable, since their electrons would radiate all of their energy as a result of constant acceleration and almost immediately fall into the nucleus.

In 1913, a Danish physicist Niels Bohr came with his own model of the atom. The Bohr model is greatly similar to the planetary model. However, Bohr specified three rules that must be strictly adhered to for the stability of atoms to be maintained:

1. Electrons orbit around the nucleus following circle-shaped orbitals without radiating light.
2. Orbitals are not located at an arbitrary distance from the nucleus, but purely on allowed energy levels that are multiples of the reduced Planck constant (reduced Planck constant has a value of the Planck constant divided by two π). From this phenomenon, it is obvious that quantization applies to objects with mass as well (in this case, electrons).
3. Electrons may jump from one orbital to another. When jumping from an orbital of lower energy to a high-energy one, an electron absorbs a quantum of light. This process is called excitation. Electrons that are located at a higher energy level than their original level are called excited electrons. In contrast, when jumping from a higher energy orbital to a lower one, an electron emits a quantum of light. Electrons that are on their original energy level are said to be in the ground state.

Using the Bohr model, the existence of the so-called spectral lines can be easily explained. A spectral line is a dark or light line disrupting an otherwise continuous electromagnetic spectrum. For example, if we expose an atom (let us consider a helium atom, for instance) to the whole spectrum, a part of this spectrum is filtered out after interacting with the atom, since certain frequencies of the spectrum have the exact amount of energy that is needed by helium electrons to move to an orbital with higher energy. Consequently, this part of the spectrum is absorbed. These disruptions of the continuous spectrum are called absorption lines. Helium electrons may never absorb the remaining radiation, because by doing so, they would find themselves outside of the allowed energy levels.

However, the radiation that was previously absorbed by the electrons is emitted after a while, when the electrons move from the orbitals with higher energy back to the ones with lower energy. Consequently, the so-called emission lines are created. Emission and absorption lines are unique for each element. This fact is used when determining the composition of remote objects in space – scientists point their telescopes at a distant cosmic body and ascertain its chemical make-up based on the spectral lines they receive.

However, even the Bohr model is not perfect and shortly after it had been published it was replaced by a more accurate model – the quantum mechanical model. Despite its imperfections, the Bohr model still presents an important transition between the classical and quantum mechanics, as it applies Planck’s findings regarding quantization to atoms.