First of all, as you can see, it is really simple. Usually, in the Physics world, simple and elegant formulas are the most important ones.
Even Einstein had this strong belief: that the world and the universe could have been described by means of some…cute formulas. A well-built theory is usually visually good in terms of equations.
As Einstein himself said: “It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.”
Just think about it. It’s like when you want to express your emotions. Sometimes you don’t think too much about them, your words come out of your mouth and you produce a messy cascade of words.
Your interlocutor can still, understand what you are saying, but it is kinda difficult to follow your thoughts. But if you write your thoughts on paper, you will probably find a better way to express them.
A more efficient and concise way. This also holds for physics theories. We could have good ones, but some of them are just more elegant than others.
Of course, this is a hard job, and our scientific inability to simplify is something we should always accept. But besides its formal beauty, the Schrödinger equation tells us something more.
It is the starting point for the understanding of quantum mechanics.
What is the Schrödinger equation?
I could start and end this explanation by saying that quantum mechanics is the study of very small things, but you would be very disappointed then. So I will try to tell you something more.
First of all, for “very small things” we mean we are interested in stuff that exists in the real world, but has atomic-scale dimensions. We are talking about atoms and subatomic particles.
So we can say quantum mechanics deals with the atomic and subatomic world. And if you take a lot of particles, you have the macroscopic world.
The macroscopic one, is the world you, me, we are used to.
In everyday life, we have to deal with macroscopic objects. Your Moka is a macroscopic object. However, it is made of atomic and subatomic particles.
Now. The macroscopic world is well described by Classical Physics, for example, Newton gave us some laws that fit well what we observe in the everyday life.
Classical physics can help us understand why the Earth orbits around the sun, why do we have seasons, how planes fly, and much more.
So it is really useful. But at a certain point, at the end of the 19th and the beginning of the 20th century, scientists realized that something was missing. When they decided to study the atomic and subatomic world, they found classical physics didn’t work anymore.
They needed another approach. This was a huge issue, a problem that needed to be solved. In fact, Physics is no more physics if it can’t describe reality.
What is Quantum Mechanics
The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the theory of relativity and the development of quantum mechanics.
The most important result is that light behaves in some aspects like particles, and in others like waves.
I know what you’re thinking: no way! And that’s pretty the same thing physicians were thinking when they first approached the undiscovered world of quantum mechanics.
Let me explain it better. Matter, the “stuff” of the universe, consists of particles such as electrons and atoms. But it also exhibits wavelike behavior. This phenomenon has been verified not only for elementary particles but also for compound particles like atoms and even molecules. For macroscopic particles, because of their extremely short wavelengths, wave properties usually cannot be detected.
Although the use of wave-particle duality has worked well in physics, the meaning or interpretation has not been satisfactorily resolved. Bohr called it the “duality paradox” and regarded it as a fundamental or metaphysical fact of nature. A given kind of quantum object will exhibit sometimes wave, sometimes a particle, character, in respectively different physical settings. He saw such duality as one aspect of the concept of complementarity.
Talking about this wave-particle duality, Einstein said: “It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.”
One of the most famous experiments that allowed scientists to understand the dual nature of matter was the double-slit experiment. It demonstrates, with unparalleled strangeness, that little particles of matter have something of a wave about them, and suggests that the very act of observing a particle has a dramatic effect on its behavior.
To start off, imagine a wall with two slits in it. Imagine throwing tennis balls at the wall. Some will bounce off the wall, but some will travel through the slits. If there’s another wall behind the first, the tennis balls that have traveled through the slits will hit it. If you mark all the spots where a ball has hit the second wall, what do you expect to see? That’s right. Two strips of marks roughly the same shape as the slits.
In the image above, the first wall is shown from the top, and the second wall is shown from the front.
Now imagine a light at a wall with two slits. As the wave passes through both slits, it essentially splits into two new waves, each spreading out from the slits.
These two waves interact with each other, and they are said to interfere with each other.
The interference could be disruptive or constructive, and in the first case, they will cancel each other out. In the second case, they will reinforce each other, giving spots with the brightest lights.
So when the light meets a second wall placed behind the first, you will see a stripy pattern, called an interference pattern.
Now, if you do the same thing with a beam of electrons, you would expect to see two rectangular strips on the second wall, as with the tennis balls, because they are particles. But what you actually see is that the spots where electrons hit replicate the interference pattern from a wave.
As you can see, this experiment suggests that what we call “particles”, such as electrons, somehow combine characteristics of particles and characteristics of waves. And this is the real essence of the quantum world.
Wave Function ψ
Basically, everything can be described or associated with a so-called wave function.
In physics, we usually indicate a wave function by means of the Greek letter “psi”: ψ
And if you remember, the psi function appears in Schrödinger’s equation.
In quantum mechanics, the Schrödinger equation is a fundamental equation that determines the temporal evolution of the state of a system, for example of a particle, an atom, or a molecule.
When it comes to quantum mechanics, intuition is no more present. For example, if you hold a ball you will notice it has some mass because of its weight.
You can feel its weight, and if you take something heavier you will notice. It is just something very intuitive, and the Classical Physics world is built upon this kind of intuition.
However, when it comes to quantum physics, things get more complex, and we soon realize we can’t predict exactly what is going to happen to the motion of a ball, for example. Or we can’t really tell, for example, if a cat is black or white. One has to imagine reality as a set of possible configurations, and you don’t know a priori which configuration is going to be chosen. We can tell that there is a certain probability associated with each configuration.
Only when we make measurements, we can see the chosen configuration, which we call state. Let’s go back to the cat. Let’s suppose you want to pet him.
We have a cat, but we don’t know if it is black or white or whatever color.
Quantum mechanics essentially states that the only way for us to know which color the cat is, is to “measure” it, and we do so by applying some mathematical objects to the “state” of the cat, which is called “operators”: you have a wave function, a state (which in this case is the color of the cat), you apply an operator, you get the result: black, white or whatever.
If you repeat the experiment a high number of times, you will end up with a distribution probability of colors, and this distribution will have a peak on one color, and that will indeed be the color of the cat you are want to pet.
One of the requirements of quantum mechanics is of course the description of macroscopic reality.
In the “big number of atoms” limit, quantum mechanics should reproduce the macroscopic world.
That is, putting in the crassest possible terms, if you see the cat is white, quantum operators applied to the color configuration should mainly give the “white” result.
Many aspects of quantum mechanics are counterintuitive and can seem paradoxical because they describe behavior quite different from that seen at larger scales.
Do you know what quantum physicists Richard Feynman said about quantum physics?
Feynman said quantum physics deals with “nature as she is – absurd”.
For example, the uncertainty principle of quantum mechanics means that the more closely one pins down one measurement (such as the position of a particle), the less accurate another complementary measurement pertaining to the same particle (such as its speed) must become.
If you know very well the position of a particle, you will lose a lot of information about its speed.
And vice-versa, you know very well the velocity of a particle, you will keep losing track of it.
Another example is entanglement, in which a measurement of any two-valued state of a particle (such as light polarized up or down) made on either of two “entangled” particles that are very far apart causes a subsequent measurement on the other particle to always be the other of the two values (such as polarized in the opposite direction).
Quantum mechanics teaches us an important thing about nature: its description is essentially probabilistic.
Before quantum mechanics, we were used to thinking that the world is governed by some given laws that give precise results. You act, you get precise results, and every action is followed by a reaction that seems to be predictable.
Instead, nature doesn’t work like that. The probability of an event—for example, where on the screen a particle shows up in the double-slit experiment—is related to the square of the absolute value of the amplitude of its wave function.
We can just say, for example, that we have an 80% of chance that we’ll find the particle in a given position interval, but we will know where it is only when we will measure it.
Another important thing quantum mechanics tells us is that is not possible to know the values of all of the properties of a system at the same time.
This is of course due to the uncertainty principle we discussed before.
Last but not least, we know that quantum mechanics is a good theory because, even if it studies the “very small things”, it closely approximates the classical description of nature in the case of large systems.