Let us start by appreciating the history first as this will be the basis of your mental picture of what quantum physics is before it gets very abstract in the mathematical structure.
Light in Newton’s days was considered to be particles, but Thomas Young with his famous double-slit experiment showed that light interferes with each other if the distance between the two slits is close to the light’s wavelength, thus light became a wave. This notion became solidified when Maxwell came out with the speed of light from the electromagnetic equations, showing that light is an electromagnetic wave, travelling at the speed of light. Thus we have the picture that electromagnetic waves unite all these radiations as one, just differing by their frequencies. From the shortest frequency to highest, we have radio waves, microwave, infrared, visible light from red to violet (following the rainbow colour arrangement), ultraviolet, X-rays and finally gamma rays. It is based on this wave theory of light which got us into the ultraviolet catastrophe.
The first sign of quantum is when Max Planck used the Planck’s constant, h to fit in the data for the black body radiation in 1900. Basically, classical theories cannot explain how light interacts with matter, predicting that as light gets to a higher frequency, and lower wavelength, there will be more ways for energy to be emitted from the matter (like when the matter is heated up). When it goes further up the ultraviolet frequency, there should be even more amount of energy emitted. This is in contrast with the experimental fact where the most common frequency of a hot body peaks depending on its temperature. Thus you see fire changes colour from red to blue as it gets hotter, and not like spontaneously releasing unlimited gamma rays. Physicists called the failure of classical theories in this area as the ultraviolet catastrophe. The X-rays and Gamma rays haven’t been discovered and named yet, or else it would be called the gamma catastrophe, which would bring about the mental image of the Hulk in most people’s mind nowadays. Maybe it is fortunate naming because this has nothing to do with the Hulk.
Planck just helped to hack the system by fitting the data in by making sure energy exchanged between light and matter happens in the form of discrete amount of energy, proportional to its frequency, linked by Planck’s constant. This is instead of splitting the energy between modes of lights which increases with the square of frequency, and allowing continuous exchange of energy between matter and light as the classical theory assumed. Planck did felt that his fitting was a mathematical trick and do not believe what the equations told him about the nature of light. That it is quantised. Hence the word quantum in quantum physics came about.
Albert Einstein then in 1905 provided the physical interpretation of this usual behaviour by suggesting that lights are particles. We call them photons. Photons as particles carry a discrete amount of energy depending on its frequency. This also explains the photoelectric effect where light only kicks out electrons from metal if its frequency goes high enough (hence enough energy per photon to kick out the electrons), regardless of its intensity (amount of photon). The electrons need a preset amount of energy to be kicked free from the metal, weak low-frequency photons can bump onto the metal all they want, but cannot combine their energy to kick out the electrons. Thus light is no longer considered as continuous wave containing continuous energy, but as photons, particles of light containing quantised energy. By the way, this is the reason Einstein got that Nobel Prize of his, not his general relativity.
This was the beginning of the crisis of interpretation.
How can a particle explain the double-slit experiment? If we assume that many photons go through the slit then maybe the particles interfere with each other. However, experiments had gone to the point where we can send individual photons to the double-slit and still after collecting enough data, the interference pattern emerges! Did the particles somehow split into two and interferes with itself? Did it interacted with a split parallel universe version of itself and recombined to form the interference? Did the particle travel through time and go through both slits at once interfere with itself and came back to the present to land on the screen? Mental pictures of the quantum world are starting to break down as we insist on using classical concepts onto the quantum particle. Weirder still, try to find out which slits did the photon goes through, then once we know which slit and cannot erase the information, the interference is gone. We get two slits of light for light going through two slits. Light behaves like a particle when information about which slit it goes through is revealed and cannot be erased away without any copies of that information. So it seems that observation changes the outcome, something totally alien to the classical world of physics where it is assumed that the observer can observe and do not affect the observed system. You might have heard of this phenomenon is called wave-particle duality. Light behaves like a wave or particle depending on our decision to observe or not to observe which path it had taken.
It seems magical now, the nature or properties of light changes depending on what we do! Some take it as there is no underlying mechanics (reality/ nature) of quantum, some disagree, this becomes a matter of interpretation. Keep in mind that the experiments and ideas which physicists came out with helped them to develop the mathematical structure of quantum theory and step by step lead them away from having a classical mental picture of reality. However, those mathematics can be used to explain and predict experimental results, because it is developed mainly to fit in with experimental results.
Next came Niels Bohr, who in 1913 introduced the atomic model which explains how atoms can be stable and the emission lines of the hydrogen atom. According to classical electromagnetic theory, if the atom is to behave like our solar system, with the nucleus of the atom in the middle like the sun and the electrons orbiting it like planets, then the electron is undergoing acceleration. Yet the electron is a charged particle, accelerating charged particle according to classical electromagnetic theory emits electromagnetic radiation. This is how radio and TV waves can be transmitted and received with the antenna. So if the electron is radiating electromagnetic waves, it must be losing energy and very soon sucked into the positively charged nucleus and the atom is destabilised. If the electrons do not move, then it will be attracted into the nucleus anyway. So it is an utter mystery how atoms which subparts of positive and negative charged particles, and the positive ones in the middle can exist at all.
Bohr suggests that electrons can only occupy some orbits, the ones which respect discrete angular momentum. Angular momentum is like momentum, spinning objects tend to remain spinning without outside forces (or torque in this case). Thus if the electrons are at the lowest orbit, it means that it cannot fall into a smaller orbit. Its angular momentum is at the lowest and cannot be reduced. There are no in-between orbits between two lowest orbits, thus angular momentum is quantised, or discretised. This, by the way, is the origin of the concept: quantum jump. As electrons cannot be found in between orbits, but jump from one to another. This is in very much contrast with our usual notion of classical motion as there is no smallest unit of jump or movement unlike in quantum systems.
In 1924, Louis de Broglie proposed that since light can behave like particles, might not particles like electron can behave like waves? The de Broglie wavelength for particles is Planck's constant over the momentum of the particle. So for very massive objects, our wavelengths are far too small for quantum effects to manifest. However, for small objects, their momentum means that their wavelength can be calculated and we can put electrons to the double-slit experiment and see that it interferes as light does. Electrons do show wave properties!
In 1925 and 1926, two different ways of getting the basic equations of quantum mechanics correct were discovered, first the matrix mechanics by Heisenberg, then the wave mechanics by Schrödinger. Both are shown to be equivalent to each other, that is different ways of expressing the same thing.
Both concepts have the concept of a state of the quantum system and an observable. The state of a quantum system is this abstract concept not directly accessible to us. What we see from experiments are the observables. Both have a system of evolution which can tell how change happens. In Heisenberg picture, the state remains constant and it is the observable that changes in time; whereas the opposite happens in the Schrödinger picture. We can call this the stage one of the quantum mechanics calculation: evolution equations. This is about the equivalent of any classical physics evolution in which time is part of the equation that tells how everything else in the equation changes or remain constant in time.
After seeing how the evolution happens, we want to know what we can observe. In classical physics, the things we can observe are obvious. Position, velocity, acceleration, force etc. Yet, state is not directly observable to us. So in quantum physics, we have to use Born's rule to translate the results of stage one of quantum mechanics to do stage two, the probabilistic part. Born's rule tells us that from the results of stage one, we can get the probability amplitude of the system. One for each possible results we can observe. Square the probability amplitude and we can get the probability density of finding each results of the experiments. And strange enough, that accurately describes all sorts of quantum experiments we care to do.
Now it is worth it to pause here and link this presentation to the usual ones you might have read in many popular physics books. If this is your first popular physics book, then just go along for the ride to recognise the terms on your second popular physics book which talks about the basic quantum theory.
Usually, the presentation uses only the Schrödinger’s picture. It's using an equation which is more familiar to physicists in the early 1900s. Wave equations. At that time, wave had united electromagnetism, optics, sound, linking to many dynamics and kinematics equations, have close relationship with the simple harmonic motion and so on. So physicists were very glad to see this familiar old friend in an unfamiliar new theory. At least for a while.
In the Schrödinger picture, quantum systems have their own wavefunction, which is the state stated above. In the Copenhagen interpretation of quantum mechanics, the wavefunction contains all possible information for whatever questions or observable you wish to ask or measure on the system. In practice, we just write the wavefunction according to the relevant observable we are interested in.
The observables can be position, momentum, energy and so on. It's the usual quantities classical physics can make sense of. So we can apply the wavefunction to the Schrödinger's equation, which roughly means how the total energy evolution of the system evolves for this particular state. The evolution here is deterministic, the same wavefunction going through the same Schrödinger's equation will yield the same resultant wavefunction to any time you care to set to. This is still stage one.
In stage two we apply the observables unto the wavefunctions to get the respective probability amplitudes for each possible results of the observable. Eg. If I want to find the position of an electron in free motion, I apply no potential energy at the Schrödinger's equation, evolve its initial wavefunction to the one I want at a certain time. Stage one completed, stage two follows. Then measure the position at that time by applying the position observable unto the wavefunction, obtaining the probability density of the position of the electrons.
If you are not mathematically inclined or had never studied quantum physics with its maths before, the above might sound gibberish to you. And it sure is very much so to many physicists in a different way. To us, we can compare it to how do you find the position of a ball in free motion. Use Newton's first law. If the ball is at rest, there is no external force on it, it remains at rest. If it is in motion, without fiction, then it will continue to be in motion.
The difference is that the evolution equation operates at stage one in quantum, a stage which is mysterious, hidden from us and all we see is the probabilistic results of stage two. There is no stage one stage two in classical physics, the evolution is clear and visible to us.
And that folks, is quantum mechanics proper. Just the maths. The story of what it means is down to the interpretations. Here lies the mystery of the quantum. Why is there two stages in the calculation? What story, if any, can we give to why is stage two probabilistic, is nature inherently non-deterministic or is it some information is hidden in stage one which we cannot know even in principle?
When Richard Feynman said, "I can safely say nobody understands quantum mechanics", he was not referring to the maths side. He is referring to the story side. With the maths side, we have the knowledge and capability to calculate and predict the probability distributions of the experimental results and so far experiments had been on the side of quantum mechanics. The calculation of molecular bonds in theoretical chemistry rely on solving super complicated equations of quantum mechanics. We can do all of these if we understand how to use the maths, even if it is super complicated.
The surprising thing is, even without knowing the underlying story of the two stages of quantum calculations, the maths still works well, predictions can be made. Nature does not seem to care if humans demand for a story.
Without that story, for you, the general layperson to predict anything in quantum systems, you would have to learn the maths. Yet, there are a few general guidelines developed in the Copenhagen interpretation, not all of which is adopted by other interpretations. Some of it you might have heard of: wave-particle duality, complementarity, superposition of states, Heisenberg uncertainty principle, inherent randomness.
We will go through them later on so as not to overly bias you towards the Copenhagen interpretation.
Why is the story important? Notice that when I used the classical ball example, I can just quote one law (Newtonian mechanics), then we can predict how the ball will behave. That's because the classical laws directly paint an obvious story for us to see and once we internalise the story, we can use it to do predictions of what will happen. In other words, it gives us power. To understand how nature works. But haven't we already know how to do predictions with quantum mechanics? What's the difference? The difference is in the intuition. The world does not behave in a quantum behaviour in our everyday experience. So as we have the intuition of how classical physics works, we would like to see if there is any underlying mechanism behind the two stages.
Brian Greene uses a theatre performance as an analogy in his book: The Fabric of Reality. In the theatre, we see the front stage, that's the probability density calculated in stage two of the quantum calculation.
Yet there is also a backstage, the place where actors change clothes really fast, where the spotlights are directed, where special effects and props are prepared, hidden until it is used. That's the stage one of the quantum calculations, the state of the quantum systems, the wavefunction. Hidden from the audiences, we do not even know to consider them real quantities in the world, or just reflections of our understanding for us to do the maths. In classical physics, the backstage is clear to us, for example, general relativity we say mass-energy curves spacetime, spacetime tells mass-energy how to move.
To make such a simple statement (or more likely, paragraphs of statements) for quantum physics means selecting one of the interpretations.
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