Quantum mechanics

Surely you have ever heard of quantum mechanics or physics. Everyone has heard this concept, but few really understand what this field of science is about.

It has been a few years since I first encountered this incredible branch of theoretical physics. I remember that it took me months to begin to understand it since some mathematical foundations of algebra and analysis are necessary to understand it in depth.

Today we are going to try to understand at a superficial level what quantum mechanics is and what interesting applications it has today.

Are you staying with us to learn a little more science?

Forward!

Quantum mechanics is a branch of physics that studies atoms and smaller particles. That is, it is the study of phenomena that occur at the atomic scales of the universe.

The quantum world is very different from what we are used to. In our daily lives we live macroscopic reality. If we see a car moving towards us, we know in a few moments it will pass in front of us. Our world is deterministic and we can explain it using classical mechanics.

In the atomic world things change a lot. An electron can be in several places at the same time and have different speeds at the same time.

These phenomena have left the best physicists in history speechless. Today we know how to work with quantum mechanics at a mathematical level and make incredible predictions. However, we are still not able to understand on a conceptual level the strange phenomena of the quantum world.

Do you want to discover more?

Stay with us and we will explain to you what the hell this quantum mechanics thing is and what applications it can have in your life!

The beginnings of quantum theory date back to the beginning of the 20th century. Its formulation occurred at the same time as the theory of relativity.

Max Planck discovered a solution to the problem of spectra of blackbody radiation. This solution was based on discrete energy values, rather than continuous ones. Here was born the first notion that the values ​​of certain observables can only take discrete values.

Max Planck hypothesized that electromagnetic radiation was absorbed by matter and then emitted in the form of packets that he called light quanta.

From here, thanks to Einstein's work on the photoelectric effect, it was seen that in certain circumstances radiation can behave like corpuscles instead of waves.

The next great step in quantum physics came from the hand of the French physicist Louis de Broglie, where he proposed an equation that related the wavelength to the mass and speed of a particle. Finally, wave-particle duality had been defined, that is, a particle had wave and particle properties.

How can this be?

Physicists didn't understand what was happening... are photons or electrons waves and particles at the same time?

What was about to come would further perplex the theoretical physicists of the time.

The great milestone of quantum mechanics was given by the great physicist Erwin Schrödinger in 1925. It was he who formulated the equation that bears his name: the Schrödinger equation.

This formula is capable of describing the evolution of a particle over time. We could say that it is analogous to Newton's equation for classical physics.

Schrödinger was able to predict the energy levels of a hydrogen atom through his equation, thus confirming its veracity.

But... What physical meaning did the wave function have?

A few years later, the physicist Max Born interpreted the square of the wave function as the probability that the particle described by said function will be found in a specific region of space.

In parallel, Werner Heisenberg formulated an alternative to Schrödinger's wave equation using matrices. This is called the matrix formulation of quantum mechanics.

Finally, Paul Dirac showed that both the wave and matrix formulations were equivalent.

It wasn't all gold that glittered. We had quantum mechanics and Einstein's special relativity. But... What happened to subatomic particles with relativistic effects?

In very heavy atoms there are many electrons around the nucleus. The repulsion between them causes them to have very high speeds, approaching that of light. At this point relativistic effects are important. For this reason we needed a formulation of relativistic quantum mechanics.

Paul Dirac solved this problem by unifying the energy equation of relativity, the famous e=mc^2, and the Schrödinger equation, obtaining what we know today as the Dirac equation.

The union of these two theories culminated in surely the most precise theories we have today: quantum field theory.

Quantum field theory is based on the study of subatomic particles and their interactions (electromagnetic interaction, weak nuclear interaction, strong nuclear interaction and gravitational interaction).

Through the quantization of classical fields and certain symmetries we can study the behavior of relativistic subatomic particles with high precision, predicting experimental values ​​with up to 11 decimal places of precision.

Don't you think it's incredible?

It is fascinating what we have achieved during the 20th century. However, there was a dream that many physicists like Einstein or Hawking had that was not fulfilled: creating a theory of everything that was capable of unifying quantum mechanics with general relativity.

There are many theories that can become the theory of everything. However, the one with the most points is superstring theory, or now known as M theory.

Simplifying, string theory tells us that all subatomic particles such as quarks or electrons are simply vibrating strings. The different modes of vibration of these strings would give rise to one type of particle or another.

What does the future hold? Will we be able to unify quantum mechanics with general relativity?

To model quantum effects, physicists used certain tools provided by mathematics. The wave function was defined as an object of an infinite vector space with certain functions (operators) that acted on them giving physical observables such as energy, linear momentum or angular momentum.

Let's go into a little more detail...

We have introduced quantum physics and its historical context. It is time to understand this concept more thoroughly and delve a little deeper into its mathematical formulation.

The wave function is a vector defined on a Hilbert space whose elements are complex functions. Furthermore, these have to be integrable square. This is essential to be able to obtain the square of the wave function which, as we have previously said, represents the probability of finding a particle in a specific area of ​​space.

It should be noted (we will see it in detail later) that the resolution of the Schrödinger equation is very complex. It is formed by differential equations with second partial derivatives that we cannot solve analytically.

There is only an exact solution for hydrogenoid atoms. These are atoms that only have one electron and one nucleus.

For polyelectronic systems we have to apply certain approximations. However, in most cases these approximations are sufficient to obtain results very close to reality.

Another essential tool of quantum theory are operators. We can define operators simply as objects that act on the wave function and return a specific magnitude.

For example, if we multiply the wave function by the energy operator, what we obtain is the energy of the particle. On the other hand, if we apply the linear momentum operator, we obtain the linear momentum of the particle described by this wave function.

Do you like math? Let's define the operators more rigorously!

An operator is a homomorphism that acts on an element of the Hilbert space and returns another object of the Hilbert vector space. In a finite vector space, for example, two-dimensional, an operator is equivalent to a 2x2 matrix.

However, in quantum mechanics we work on infinite Hilbert vector spaces, therefore, the matrix associated with this endomorphism would be of infinite dimensions. For this reason, we work with operators.

Working with these operators we can obtain all the properties of any system that interests us.

Mathematically we can work without problem with all quantum phenomena. We are able to calculate and predict many experimental results. However there is a problem…

We are not able to understand what happens in the quantum world on a conceptual level.

An experiment that captures this fact very well is the double slit experiment.

In general terms, this experiment tells us that if we measure the state of a system, it behaves like a corpuscular particle. However, if we do not measure any property of it, then it behaves like a wave. So does the particle know that someone or something is measuring it?

This is our problem... How can the particle know this?

So if the very act of measuring changes reality, we will never know what is really happening because by observing reality we modify it.

There are different interpretations to try to explain these rare events that occur in the smallest world.

This interpretation, which describes quantum mechanics as a correct and complete theory, was proposed by Niels Bohr, Max Born and Werner Heisenberg and explains that when the particle is not observed it evolves in time following the behavior described by the wave function.

Furthermore, this mathematical object, the wave function, implicitly contains all the information of the system under study.

The square of the wave function describes a probability function that tells us how likely it is to find the study particle in a given space.

Finally, when this system is measured, the wave function collapses into one and only one of the possible states marked by the probability distribution postulated by Max Born.

The interpretation proposed by Hugh Everett says that reality doubles in the different possibilities that a system can adapt. It is as if it branches out and creates different parallel realities. Therefore, there is no longer a collapse of the wave function, but rather all possible realities take place.

The theory of hidden variables postulates that there are certain unknown parameters that are responsible for the stochastic or probabilistic effects of quantum theory.

Therefore, all the inexplicable phenomena that occur in the subatomic world, such as the superposition of states or the impossibility of accurately measuring position and velocity, would be due to these hidden variables.

If we knew them, quantum mechanics would act like classical statistical mechanics. Then it would become a deterministic theory and totally analogous to classical physics.

Quantum physics is here to stay and more and more applications are emerging in our society. Some fields where technologies based on quantum theories are used are biology, medicine, communication sciences or quantum computing.

Furthermore, it is essential in the study of fundamental physics that allows us to understand the laws of the universe and how it works.

Quantum chemistry is used in fields such as computational biology or materials chemistry.

Many drugs today are designed through computational protocols. What is done is to model the target protein and see how it interacts with different molecules to determine which of them is suitable to act as a drug.

The interaction between the protein and the drug occurs through intermolecular forces that can be explained precisely thanks to quantum chemistry, solving the Schrödinger equation. In these cases, the system associated with the Schrödinger equation contains many electrons, therefore, the solution must be obtained approximately.

Quantum can also be used in the biotechnology industry. Many enzymes used in cleaning products or for the food industry can be optimized to improve their performance and lower production costs.

To carry out this task, the chemical reaction that takes place within the protein must be computationally modeled. A chemical reaction is nothing more than an exchange and rearrangement of electrons, therefore, to model it you need to use the Schrödinger equation.

We have all heard about the benefits that quantum computing will bring to our lives. Quantum computers open a new paradigm in computing and will allow us to make calculations that until now we thought were impossible.

I have to tell you that it is not entirely like that…

Computers with quantum technology will be a complement to current processors. In many situations it is not necessary to use quantum processors since they provide additional calculation speed.

They are in certain types of algorithms in which quantum ones perform much better than classical ones, but not in all.

However, quantum computers have many applications. For example, drug simulation could advance greatly thanks to quantum technology since we could make much more precise simulations and greatly improve the computational design of medicines.

He also has a keen interest in the field of cybersecurity. Quantum computing could break some of the encryption systems that exist today. But it would also help create encrypted protection systems that are much more secure and difficult to break.

Another interesting area for the application of these new processors is in machine learning or artificial intelligence. There are certain algorithms in this field that could be improved with this new technology, achieving very impressive results in artificial intelligence.