You're asking a tough question! I try to explain QM to nonphysicists all the time, and it's really hard; here are a few of the accessible explanations I've found.

What I'll say below is not logically airtight and is even circular. This is actually a good thing: motivation for a physical theory must be circular, because the real justification is from experiment. If we could prove a theory right by just talking about it, we wouldn't have to do experiments. Similarly, you can never prove to a QM skeptic that QM must be right, you can only show them how QM is the simplest way to explain the data.

Quantization $rightarrow$ Superposition

If I had to pick one thing that led to as much of QM as possible, I'd say quantum particles can exist in a superposition of the configurations that classical particles can have. But this is a profoundly unintuitive statement.

It's easier to start with photons, because they correspond classically to a field. It's intuitive that fields can superpose; for example the sound waves of two musicians playing at once superpose, adding onto each other. When a light wave hits a partially transparent mirror, it turns into a superposition of a transmitted and reflected wave, each with half the energy.

We know experimentally that light comes in little bullets called photons. So what is the state of a photon after hitting a partially transparent mirror? There are a few possibilities.

1. The photon is split in half, with one half transmitted. This is wrong, because we observe all photons to have energy $E = hf$, never $E = hf/2$.


2. The photon goes one way or the other. This is wrong, because we can recombine the two beams into one with a second partially transparent mirror (the would-be second beam destructively interferes). This could not happen if the photon merely bounced probabilistically; you would always get two beams.


3. The photon goes one way or the other, but it interferes with other photons somehow. This is wrong because the effect above persists even with one photon in the apparatus at a time.


4. The photon is something else entirely: in a superposition of the two states. It's like how you can have a superposition of waves, and it is not a logical "AND" or "OR".

Possibility (4) is the simplest that explains the data. That is, experiment tells us that we should allow particles to be in superpositions of states which are classically incompatible. Then you can extend the same reasoning to "matter" particles like electrons, by the logic that everything in the universe should play by the same rules. (Of course such arguments are much easier in hindsight and the real justification is decades of experimental confirmation.)

Superposition $rightarrow$ Probability

Allowing superposition quickly brings in probability. Suppose we measure the position of the photon after it hits one of these partially transparent mirrors. Its state is a superposition of the two possibilities, but you only ever see one or the other -- so which you see must be probabilistically determined.

This is not a proof. It just shows that probabilistic measurement is the simplest way to explain what's going on. You can get rid of the probability with alternative formulations of QM, where the photon has an extra tag on it called a hidden variable telling it where it "really" is, but you really have to work at it. Such formulations are universally more complicated.

Superposition $rightarrow$ Entanglement

This is an easy one. Just consider two particles, which can each have either spin up $|uparrow rangle$ or spin down $|downarrowrangle$ classically. Then the joint state of the particles, classically, is $|uparrow uparrow rangle$, $|uparrow downarrow rangle$, $|downarrow uparrow rangle$, or $|downarrow downarrow rangle$. By the superposition principle, the quantum state may be a superposition of these four states. But this immediately allows entanglement; for instance the state
$$|uparrow uparrow rangle + |downarrow downarrow rangle$$
is entangled. The state of each individual particle is not defined, yet measurements of the two are correlated.

Complex Numbers

This is easier if you're speaking to an engineer. When we deal with classical waves, it's very convenient to "complexify", turning $cos(omega t)$ to $e^i omega t$, and using tools like the complex Fourier transform. Here the complex phase is just a "clock" that keeps track of the wave's phase. It's trivial to rewrite QM without complex numbers, by just expanding all of them as two real numbers, as argued here. Interference does not depend on complex numbers.

Wave Mechanics

The second crucial fact about quantum mechanics is momentum is the gradient of phase, which by special relativity means that energy is the rate of change of phase. This can be motivated by classical mechanics, appearing explicitly in the Hamilton-Jacobi equation, but I don't know how to motivate it without any math. In any case, these give you the de Broglie relations
$$E = hbar omega, quad p = hbar k$$
which are all we'll need below. This postulate plus the superposition principle gives all of basic QM.

Superposition $rightarrow$ Uncertainty

The uncertainty principle arises because some sets of questions can't have definite answers at once. This arises even classically. For instance, the questions "are you moving north or east" and "are you moving northeast or southeast" do not have definite answers. If you were moving northeast, then your velocity is a superposition of north and east, so the first question doesn't have a definite answer. I go into more detail about this here.

The Heisenberg uncertainty principle is the specific application of this to position and momentum, and it follows because states with definite position (i.e. those that answer the question "are you here or there") are not states of definite momentum ("are you moving left or right"). As we said above, momentum is associated with the gradient of the wavefunction's phase; when a particle moves, it "corkscrews" along the direction of its phase like a rotating barber pole. So the position-definite states look like spikes, while the momentum-definite states look like infinite corkscrews. You simply can't be both.

Long ago, this was understood using the idea that "measurement disturbs the system". The point is that in classical mechanics, you can always make harmless measurements by measuring more gently. But in quantum mechanics, there really is a minimum scale; you can't measure with light "more gently" because you can't make photons smaller than they are. Then you can show that a position measurement with a photon inevitably changes the momentum to preserve the Heisenberg uncertainty principle. However, I don't like this argument because the uncertainty is really inherent to the states themselves, not to the way we measure them. As long as QM holds, it cannot be improved by better measurement technology.

Superposition $rightarrow$ Quantization

Quantization is easy to understand for classical sound waves: a plucked string can only make discrete frequencies. It is not a property of QM, but rather a property of all confined waves.

If you accept that the quantum state is a superposition of classical position states and is hence described by a wavefunction $psi(mathbfx)$, and that this wavefunction obeys a reasonable equation, then it's inevitable that you get discrete "frequencies" for the same reason: you need to fit an integer number of oscillations on your string (or around the atom, etc.). This yields discrete energies by $E = hbar omega$.

From here follows the quantization of atomic energy levels, the lack of quantization of energy levels for a free particle, as well as the quantization of particle number in quantum field theory that allow us to talk about particles at all. That takes us full circle.

I'm no physics grad student, but I've had to come to grips with similar oddities in QM just from my own hobby perspective. If I were to explain QM to a complete newcommer, to try to get them comfortable with the quirks, I'd start with what is probably the single most important aspect of QM. Like all other scientific theories, it is a model. It is not reality, it is a description of reality. It's just a description that's so good that it can be difficult to tell the difference.

If they can grasp that, the next step is to point out that the observations which lead to the development of QM are simply pesky. They refuse to fit together in any simple solution. The whole reason we have QM is because we started exploring corner cases, and our existing models fell apart. So when the QM models say something counterintuitive (read: entanglement), it's because there were some really pesky experiments done which showed that yes, this is the way our universe really works (Such as those demonstrating the Bell inequalities)

At this point, I consider wave-particle duality to be an essential next step. You mention not wanting to dig into it, but when explained properly, I think it adds to clarity rather than murkiness. The key is to explain that there used to be two models for how light worked. One used math associated with particles, and one used math associated with waves. We often say "light is a wave and a particle at the same time," but that's the confusing phrase. I prefer to say "light is neither an EM wave nor a particle. It is something which sometimes behaves really similar to a wave, or really similar to a particle, but when we put it into exotic situations, it behaves as something completely different. (That "something completely different" happens to be well-modeled using superposition)

At that point, if they are comfortable with this slight twist to normal terminology, you can start going into the findings and the thought experiments and so forth. I find it is far easier to explain Schrodinger's cat as "the cat is neither alive nor dead, it's something else" than it is to say "it is alive and dead," which is designed to fly in the face of their understanding.

In the end, what makes QM hard is not the QM. As you said, QM is easy. What makes it hard is that the models QM use suggest that a lot of things we take for granted on a daily basis aren't quite accurate. If one is gentle with how one unsettles all of these assumptions that have been made over years, it's much easier to get someone to be comfortable with QM.

Consider this: using the classical model of the atom, your hand and the desk are well over 99.99% empty space. It's merely electrostatic repulsions that give the perception of these objects being solid. You and I know this. But most people are not inuitively comfortable with it. It bothers them to think that objects are that ephemeral. Now if that causes cognative dissonance in someone, is it not reasonable that they would have cognative dissonance with QM, which suggests that the concept of something being empty space isn't even really meaningful?

The theory of quantum mechanics became necessary when physical measurements could not be fitted within the mathematical frameworks of classical mechanics , electrodynamics and thermodynamics.

AFAIK three experiments were fundamentally not fitting the classical models.

1) Black body radiation

2)the photoelectric effect

3) the atomic and molecular spectra , showing discrete lines.

At a lower level, the existence of atoms themselves, electrons orbiting around a positive nucleus in no way could be explained in a stable classical model, as eventually the electrons should fall on the nucleus and become one. ( I have an answer here to this which is relevant to this answer)

The Bohr model showed the way to mathematically model 3) , Planck's assumption of quantization in photon energies in the black body solved the ultraviolet catastrophy, number 2), and the photoelectric effect was explained by the assumption of discrete energies needed from the photons to release bound electrons from surfaces.

The theory of quantum mechanics tied in a mathematical model based on specific wave differential equations and a number of axioms (postulates) , explained existing data and predict new set ups.

I am trying to say that quantum mechanics is not a brilliant mathematical model somebody invented, but a mathematical description of nature in the microcosm forced on us by data.

The probabilistic nature, was forced by the fits to the data.

I'm not interested in why we need QM, I want to know/explain what fundamental assumption is necessary to build the theory.

The fundamental assumption is that a physics theory had to describe/fit existing data, that the classical theories could not, and that it should successfully predict new setups.

The fundamental assumptions/axioms are given in the postulates of quantum mechanics, so as to pick from the solutions of the appropriate differential equations, those functions that fulfill the postulates.

All your list is the mathematical consequence of the above choice, and I do not think that it something simple to explain to non mathematically sophisticated people. Complex numbers was a separate mathematical course in my time ( 1960s)

Edit after edit of question:

what QM is about, what happens in the "quantum world", what phenomena it correctly predicts, no matter how counterintuitive to our classical minds.

To introduce somebody with minimal mathematics, to the quantum mechanical framework, I would state that "there are no absolute dynamical predictions for the behavior of matter. Quantum mechanics is all about probabilities, not certainties"

In classical mechanics there is a simple equation when a ball is thrown that describes its trajectory. In quantum mechanics the microscopic trajectory of a particle is not predictable, but probable: there are many paths that the particle could take at the nanometer and smaller scales.

I would then give them the double slit interference experiment one electron at a time. The experiment is "electron scattering off two slits with specific width and distance apart", which creates the microscopic quantum mechanical framework, and the electron is detected macroscopically at the screen The accumulation of electrons shows up that the probability distribution didsplays wave interference patterns, while the point on the screen , within nanometers, has a specific (x,y) characteristic of a classical particle footprint. This demonstrates the wave particle duality.

Once the wave nature is demonstrated, the existence of phases in the description of waves allows to explain superposition and entanglement with simple analogies to water waves and pendulums BUT holding for probability distributions.

A minimum of mathematics is necessary, but mathematics is also necessary to explain classical physics behavior too.

I'm not a grad student in Physics, but I have an undergrad degree in it, so I guess I'm a potential grad student in Physics.

I've had >30 years to reflect on it, and when people ask me about it, I usually give them an answer something like this:

Quantum mechanics is a collection of ideas that work together to allow Physicists to make sense out of some otherwise incompatible-seeming laws of Nature. It makes the Math come out right, but leads to mostly "non-intuitive" conclusions. Einstein never "bought" it, and came up with a couple of the most common insults to QM. But literally the rest of the Physics community did "buy in" to QM, because it makes the Math work out. Personally, I hope it eventually proves to be a stopgap, and we'll find the underlying principle that makes QM make sense, but we're not there yet. (In that regard, it's very much like "dark matter".)

I think this explains the "situation" of QM better than any inherently-confusing "explanation" of a collection of seemingly-unrelated ideas, and introduces the idea that Physics is messy.

P.S. I count "playing with dice" and "spooky action at a distance" as Einstein's "insults". Just like "Big Bang", the latter has moved away from being an insult, but was originally intended as one.

Edit: I imagine some (many?) will complain that this isn't an "answer". To the degree that the question is valid, that's true. But I don't think the question really is: it presupposes that there is some central organizing principle (or two) behind QM. From the length of the high-quality "answers" we've seen so far, I think it's pretty clear that QM just isn't there (yet).

Of course you see Anna V's answer gives you why. You don't want why, I understand.

You want to explain what is QM. You want to explain to everyday people what QM is.

1. Quantization of energy

In this case, what is QM will partially be answered by Anna V's answer to why. In the old times, we only saw the world at a macro level, and did not know how micro level worked. We built up our mathematical system to describe the world visible at the macro level. This mostly worked. Then came a few problems, like those three mentioned, and we had to build (and use for physics) a new mathematical system that described the micro level. And so you had the classical standard system describing the macro level (SR, GR) and the new system (QM) describing the micro level. The two are not always compatible. This answers your questions to the quantization of energy. You can explain to everyday people, what QM is in terms of how it describes quantization of energy. That is, how it solved those three problems mentioned.

2. Interference

But I understand from your comments you need superposition and interference explained.

Interference is easiest explained to an everyday person with the double slit experiment. It is as simple, as saying, that the photon travels as a wave, and the parts of the wave will go through both slits, and after the slits, the parts of the wave will interfere with each other, creating interference. The dark parts on the screen are the destructive interferences, and the bright parts are the constructive interferences. This is understandable to an everyday person. Even if you do it with one single photon at a time, that one single photon, traveling as a wave will interfere with parts of its own wave. You cannot know where the photon will land on the screen exactly, but you can know where it will not land, on the dark parts of the pattern. And you can know upfront, based on the wavefunction (which is just the probability distribution of the particle's position for everyday people) of the photon, and the slits, where the bright and dark patterns will be.

3. Superposition:

This is best understandable for an everyday person from the neutrino experiments. Neutrinos are traveling in a superposition of three flavors (that is three masses for everyday people). Because each of them has different masses, and energies, they too travel at different speeds. By the time they arrive, they will seem to be only of one certain flavor. This is because the oscillation is periodic, the neutrino will return to the original flavour after a certain distance traveled. If they know the distance traveled, they know which flavor it has oscillated to when arriving, so they will know which flavor it started with when created.

4. Uncertainty:

This is the most important to explain to an everyday person. You have to accept that we are checking micro level objects at scales, that we really do not have a good way (with little error) to look at. In the old days, when we were checking things, that is macro objects, they were spatially stable. Even at smaller scales, we had something that was way faster then the object that was being checked. That was the electron microscope, and that was the last scale that we could take a pretty good look at objects at small scales (atoms, protons). Can we do that with another electron? Can we create a classical picture of an electron? Or a photon? Or a quark? Or a gluon? No. Why? There are a few reasons. First, they are accepted to be pointlike particles. Is there anything to check them with (what they would look like classically)? No there is not. How can you check a pointlike something what it would look like in a classical way (because an everyday person wants to see most importantly what an object looks like classically)? You can't. Second, they move too fast to be checked. Micro objects most of the time tend to move close to the speed of light. To check something in a classical way, you need to check them with something that moves way faster then the object itself. The classical way to check something, to an everyday person, is to look at it. You use EM waves. And your eyes. Can you do that if the object moves near the speed of light? Could you see (and check thoroughly) a macro object moving near the speed of light? Not really. This way you can explain to an everyday person what QM is, in terms of uncertainty. Too small things (pointlike) move too fast (close to speed of light). The more we try to classically confine them to a smaller place to check them, the more they will speed up (their momentum will be uncertain). So to describe them, we need probability.

Entanglement:

This is something that you cannot explain to an everyday person. This is the part of QM, where nobody really can explain, and there is no accepted answer to the math and the experiments. The math describes the experiments, but nobody can explain classically why. It is called correlation. There is correlation between certain characteristics of entangled particles. There is the coppenhagen and the many worlds interpretation. Whichever you like more, the everyday person will have to choose.

SR:

I see that you edited your question, and wonder why and how to explain to an everyday person that speed of light is the same from every frame. The way to explain that to an everyday person is to explain to them that they try to think the wrong way. You think you could speed up to the speed of light. Wrong. Energy, and GWs travel at the speed of light. That is the speed, the only speed of energy. To slow down in space you need to gain rest mass. Everything is relative to the speed of light, c (that is the speed of EM waves and GWs). Once you accept that the universe is set up so that these waves travel in vacuum (when measured locally) at the speed of light, and everything that has rest mass slowed down compared to that, you will see that there is no contradiction in your train example. You and the train and whatever you throw away travels at speeds 0.000c, 0.0001c etc. In that case, c will always seem to be c from everybody's point of view from every frame. That is made so that we, who have rest mass can experience time, and start moving in the time dimension (photons and gravitons do not experience time as we do).

I'm just going to give an answer here but I've thought about a bit. In particular, previously I prepared a intro presentation on quantum mechanics which I've given to high school students a few times and I've been happy with the results in terms of getting their feedback and getting them to follow the discussion and ask questions.

Like you I've been dissatisfied with many of the introduction to quantum mechanics I've come across. Things like wave-particle duality and the double slit or stern-gerlach experiment were helpful for physicists in the early 20th century because those experiments were in the realm of types of experiments those scientists were used to thinking about so the results were very striking for these scientists. Young physicists and lay people now aren't so familiar with those apparatuses so it the results are not so striking.

I think there is another important difference between teaching quantum mechanics now and teaching quantum mechanics 70 years ago. The difference is that listeners are VERY ready to accept crazy things coming out of the mouth of a physicists. "Oh the physicist says the electron can be in two places at once and in 6 dimensions? Yeah they're probably right and it's just something I'll never understand." I learned about the quantization of atomic orbitals probably before high school so I was not surprised about it when I learned the mathematics of the quantization in college.

I think the most important thing with introducing quantum mechanics is to get people to feel that what they are being told is really weird. If they don't feel it's weird you haven't succeeded.

Anyways, my approach goes like this.

1) First I remind people that Quantum mechanics is a subfield of physics which is a type of science. The work of science is to come up with, and test, theoretical models which best describe/predict our experience in this world (namely describe/predict experiments we can perform). This setup is good because it allows you sidestep questions like: "but is the cat REALLY dead and alive?" by allowing listeners to recall that science doesn't necessarily tell us what is TRUE but it does give us the best DESCRIPTION of reality to date. the interpretation of the physical model is a different (but related/important) part of the story.

2) This is the key to my approach. You mention that you don't want to get into a discussion of postulates/Hilbert space etc. I agree, but for me it is the Hilbert space which gives us the main difference between classical and quantum mechanics so I can't let this slip by. The way I introduce this to a lay person is this. First never ever mention the word Hilbert space. That would be terrible. What I do is first explain the idea of the state of a physical system in a classical sense. I define the state of a system as being a collection (like a bag) of descriptions of a physical system. The physical state of a water bottle could be: "Blue, 1 Kg, moving at velocity v downards, just above the table, liquid state, not rotating" etc etc. I then make the very simple and believable statement that when you zoom in on smaller and smaller physical systems (atoms/molecules) the state of the system becomes more complicated. Namely, we need to add possibilities to what sorts of states the system can be in. As a teaser, classically a ball can't be in two places at once, but quantum mechanically it can.

3) The next bit of the description is to start to go into examples of how specifically the state of the system can differ. For me the important differences are the possibility of superposition and entanglement. However we can add from the OP's list interference and the uncertainty principle. In my presentation I usually only go through superposition, I also try to avoid the much used physics examples like Schrodinger's cat because people have maybe already heard that before and gotten the sense that only cats can be in superpositions. To explain superposition I lean on my description of a physical state. I say to think of a classical state like a bag of description. Quantum states are also bags of descriptions. The interesting thing is a quantum state bag can include MULTIPLE different classical state bags inside of it. So if a four-square ball can be in one of four boxes classically, quantum mechanically the description can simultaneously include parts where the ball is in any one of those boxes.

4) At this point I address common misconceptions about what superposition might be. I consider a dog which is in a superposition of being a red dog and a blue dog. Possible misconceptions are: Half the dog is blue and half the dog is red, the dog is purple, the dog switches back and forth really fast between being red and being blue, the dog is really only red or blue. I think helps to explicitly point out these misconceptions because whoever you are thinking to is probably thinking that one of these is correct. It's ok and important with quantum mechanics to tell someone they're picturing it wrong even if you can't immediately give them a not wrong picutre. And this is my next point.

5) Next, and this is where I really differ from many introduction I have seen before: I introduce different possible interpretations of superposition/quantum mechanics. Many physicists seem to think interpretations are taboo and shouldn't be discussed or "aren't physics". Ultimately I disagree with this.

I start off by asking Schrodinger's obvious question. If we put something in a superposition then what happens when we look at it? What if we start with a regular dog and we have a machine that can put it in a superposition of being red and blue. What would happen when we look at the dog? I first point out that no one in the world truly knows the answer to this question, namely, the measurement problem is still unsolved. This put into perspective the weirdness level*. I give the description that three interpretations of quantum mechanics would give of this experiment. Copenhagen interpretation, many worlds interpretation, and hidden variables interpretation. I then try to explain pros and cons of each interpretation. I think this is good because it shows people that yes, quantum mechanics is weird, but that there are few different ways they can think about it. I think for me, it is important to remember that a) these different interpretations can all work, but b) they do all have problems.

While giving this presentations I've got many interesting questions and had interesting discussions. I've brought up (but always used non-expert language) entanglement, Rabi flopping, an egg rolling off the top of a barn (symmetry breaking) and other interesting topics. I have also come up with some other interesting ways to explain entanglement.

The closest everyday analogy I've come up with for superposition is the idea of travelling Northwest. You are travelling north, but you are also travelling west. Both statements are simultaneously true. You're also doing something slightly different than going purely north or purely south. This captures the idea that Hilbert space is a vector space just like the cardinal directions are vectors.

Another superposition which is interesting is a particle in a superposition of travelling left and travelling right. Classical, the two vector would cancel out and we would see not motion. But quantum mechanically both things can be happening at once

*In hindsight I don't want to give the impression that quantum mechanics is so weird no one understands it. I only want to point out that the philosophicalish measurement problem is unsolved. I think it is important to impress upon people that we in fact understand quantum mechanics very well and can use it to make incredibly precise prediction as well as use its results to build real technology that helps everyone in their every day lives.

When they [(non-physicists) friends and family] ask me "So what is QM all about?", I'd like to give them a short answer, a simple ...

You have no chance at all to explain QM in a few words to those people. The best you can do is give them a very broad overview of the purpose or realm of QM, the motivation why we as humans study it, maybe emotions you feel while working with it, etc.

"You know how the greeks thought that there must be atoms - the smallest bits of matter that cannot be subdivided anymore. They were right; today we know that atoms are in some fashion the smallest pieces of matter. But, as you know from school, even atoms are immediately broken down into electrons, protons and neutrons. This also, in school, happens pretty simply; you know how the electron fizzes around the proton-neutron core like small planets or billiard balls. This description of atoms is enough to handle most of chemistry and physics as we need it for practical applications, for example to create cool new materials like special lightweight alloys for airplanes, do understand how gas or water works, to shoot rockets, and so forth."

"Now, in the 20th century we found that neutrons, protons, electrons and even light can be split into even smaller pieces, or at least explained more thoroughly; and that's what QM is all about. Those inner workings are completely fascinating, but also very unimaginable and there is no way to find a simple image like billiard balls or planets for them. I'm afraid physicist, even ones better than myself, don't really have good descriptions here, but need hours and hours of mathematical studies to get a slight grasp on it. We also don't have real practical uses of it yet, but it is one of the borders of knowledge that we are always expanding..."

Source: am not a physicist, but an IT / CS guy. I have been asked to explain things like "internet", "computer", "algorithm", "A.I." etc. to non-technologists. I soon turned to very very simple images and skipping any level of technical detail after being met with glossy eyes many times when explaining these - compared to QM - extremely simple things. There certainly is no way at all that can you give a sufficient impression of QM to people not used to advanced physics ideas.

Even people who were arguably pretty good at this do not give a short answer what QM is about. Feynman, in the many videos which exist of him, regularly needs hours to talk about QM or other stuff, with an audience which is probably largely physicists. He never starts his talks with a small explanation in a few words, because he deems this quite impossible (i.e., Feynman - Why.) from an ontological point of view.

I've had to answer this in relation to explaining Schrödinger's Cat to a non-physicist who wanted a basic understanding of what it was about. I covered it roughly like this:

1. We now know that commonsense certainties and classical rules/"laws of nature" break down on the tiny scale of atoms: - Subatomic particle behaviour is extremely non-intuitive.

• we have nice solid rules governing the visible universe. Newton and Einstein for gravity and forces, Maxwell's laws for electricity and magnetism, and so on.




• But we now know that as you go smaller in scale, towards atoms and subatomic particles, they actually don't behave that way. What we are actually seeing is a kind of "averaging" or statistical effect due to billions and quadrillions of tiny particles.




• For example (and simplifying!) on average, electrons flow within a wire causing electric current. But a single electron could do any number of things. The current is predictable because across a billion billion electrons, we will see fairly predictable average behaviour overall. But the behaviour of an individual electron is not predecided that way.




• So when we look at the "building blocks" of the physical world, we have to work with particles that dong behave as we expect. They are quite likely to do this, a bit likely to do that, and extremely unlikely to do some 3rd thing (but will do it once in a million years or whatever). And we actually cannot predict which they will do.




• Quantum mechanics is the term used to describe the laws that seem to describe how the smallest physical parts of our universe work,and how on average they create the apparently predictable laws we "see" around us in everyday life.

2. What it implies

• The fact that the world of subatomic particles works this way, means it has a lot of surprises for us, that just don't make sense on a "day to day" level.




• But they do seem to be true, even if they don't make sense, we have tested them for about a century, we use them to build computers and lasers that wouldn't work if this was very wrong. So we know that however weird it seems, it does seem this is how the world is.

Examples:

1. For example, two golf balls cannot occupy the same space. But two of some kinds of subatomic particles can.




2. An object like a chair is in a specific state - it exists or it doesn't. But a subatomic particle can exist-somewhat - or both exist-and-not-exist, or maybe-exist, which makes no sense in the everyday world.




3. In the everyday world, effects almost always have identifiable causes. If a box suddenly moves, something made it move - a chain of cause-effect events. In the quantum world, things can happen at random, without something else "causing" them.




4. We can measure how fast a car is going, and where it is. But in the subatomic world, as soon as you try to accurately measure one of these, the laws of nature stop you from accurately measuring the other.




5. A golf ball rolled towards a set of widely spaced railings, will go through one space. But in the quantum world the same single particle (or wave) can go through two different spaces, at the same time - and "bounce off" (interact with) itself on the other side.

3. Where this gets us

• This is not very intuitive, and don't ask me to explain how it can be. But it is what we find happens, and we are pretty sure of it.




• Schrödinger's Cat is a way to try and show by an example, just how strange it is, by considering a setup in which a (hidden) cat could be alive or dead, but we don't know which, and its state is controlled by the behaviour ofa single subatomic particle. Because the subatomic particle's state is uncertain, we actually don't know the cat's state either.




• Quantum mechanics says that - as best we understand it - the right way to describe the cats state is that it is both alive-and-dead when we can't see it - and when it is revealed, one or the other states will become "our reality".




• And if that sounds weird and impossible - I agree!!

Quantum mechanics, at its heart, is a framework for describing the behaviour of the world in certain conditions, one that we need to accurately describe certain parts of reality. That's all there is to it. There is nothing inherently strange about quantum mechanics, only the fact that it apparently contradicts much of our everyday intuition, and indeed many concepts from classical mechanics we thought were fundamental. When we think about it, this is to be expected. After all, we are not attempting to describe everyday phenomena at our scale, that we're familiar with, like a stone being thrown, or a tree swaying in the wind: phenomena for which our brain has evolved to process. When attempting to make sense of quantum mechanics, never forget this: our brain technically wasn't meant for this! It was meant to locate tigers in the jungle, or dodge or catch things flying through the air. Don't feel bad if these concepts (though accurate) seem unintuitive or even weird to you. It's only natural!

I will now attempt to succinctly explain the "core" of quantum mechanics, and why it follows from the experimental data (the starting point for any physics).

(The following summary is inspired by Ch. 1 of Cohen-Tannoudji)

It all starts with the double slit experiment. This is our setup: shine a light source onto an opaque plate, which has two narrow slits. Behind it, a screen lets us see the resulting light that shines upon it, and observe any interference patterns that may arise.

Before we go further, let's briefly recall the two main ideas to describe light (before quantum mechanics): "light is a particle", and "light is a wave". Looking at things like the photoelectric effect, and blackbody radiation, and the simple fact that light is emitted in discrete entity (it's never possible to emit light with energy less than $hf$, we may be led to conclude that light is a particle. Looking at things like diffraction and the fact that light appears to be a type of electromagnetic wave, we may be led to conclude that light is a wave. So, which one is it?

Let's look at our experiment. What we observe is this: shine a light onto one of the slits only. Call the observed pattern (of light intensity) $I_1(x)$. Now shine it onto the other slit, and call the pattern $I_2(x)$. Great! Now, shine the light on both slits, and call the observed pattern $I(x)$. If light were a particle, you would expect $I(x)$ to simply be $=I_1(x)+I_2(x)$, but this is not the case. You could attempt to explain it in the framework of particle theory by saying that the particles exiting both slits interfere with each other, causing the diffraction pattern for both slits to be different from the sum of the two separate patterns of the two slits. Let's hold that thought for a moment.

Now, let's try "reducing" the intensity of the light source, $I_textsource$. If light is not a particle, but a wave, as the previous observations seem to imply, then we simply expect that the intensity of the diffraction pattern is going to be proportional to the intensity of the source: turn the source halfway down and the diffraction pattern will be half dimmed, turn it to 1/10th and the pattern will be 1/10th of the intensity, until you turn it off and the pattern disappears.

This, in fact, is NOT what happens. Let's turn down the source, in fact let's turn it way down such that it emits only one photon (minimal light quanta) at a time. This is where the magic happens! Neither "light is a particle", nor "light is a wave" are sufficient descriptions of what happens!

1. Leave it on for the night: when you come back, the diffraction pattern is there, even though only one photon was emitted at a time, such that particle interactions between them, as we proposed earlier, cannot happen! The "light is a particle" explanation is definitely ruled out.




2. On the other hand, look at the screen after each photon is emitted: you will not find, as we speculated, a very very dim diffraction pattern. Surprisingly, unbelievably, we will find discrete hits! The "light is a wave" explanation is definitely ruled out, since one photon emitted corresponds to one hit on the screen, rather than very dim pattern.

So, if "particle" and "wave" aren't good descriptions of light, what is? The answer necessarily must be: a new class of object, which is neither a particle nor a wave, but a completely new concept: a quantum particle. A wave-function, that contains all the information there is about the particle, and a mechanism that describes how our "observables" (position, momentum, spin, polarization, etc) relate to that wave-function. All of the "weird" behaviour follows from there. It neither conforms to our intuitions of how particles or waves behave.

This is only the starting point. I you want to learn how the rest of quantum mechanics follows from here and truly understand the physical aspects of QM, I heartily recommend chapter 1 of "Quantum Mechanics" by Cohen-Tannoudji et al.

I've never tried to explain all your points, but the core idea I give is "A state might not determine a measurement outcome".

I first explain the idea of a physical state - that a physical system can be in one state or another. And before QM, the sate was always thought to determine the measurement. If we say that the ball's velocity is v, then this means that there is a state (which we can even mark as $|vrangle$, I don't think this hurts the explanation) so that if we measure the ball's velocity we'll find that it's v. This bit goes down well, it's an easy idea to comprehend.

Then I state that the amazing idea that QM is based on is that there are states that do not determine a certain measurement. So there is a state, let's call is $|strangerangle$ where if I measure I get either a certain velocity $v_1$ or some other velocity $v_2$.

Quantum superposition and indeterminism

This naturally leads to a discussion of quantum superposition and indeterminism. I explain that we write these strange states as sums of the regular states, and QM states that such sums are always legitimate states. So if the particle can have a state $|v_1rangle$ (meaning we'll measure velocity $|v_1rangle$) and a state $|v_2rangle$, then it can also have a state $|v_1rangle+|v_2rangle$, which means that upon measuring it we'll measure either velocity.

This means that we need to measure either velocity with some probability. And indeed we can make a weighted-sum to alter the probability which-way. For example the state $sqrt1/3 |v_1rangle+sqrt2/3|v_2rangle$ means that $|v_2rangle$ is twice more probable. I might even explain that for technical reasons the square root of the probability is what we need to put in front of the classical-state.

Entanglement

That's usually as far as I go. But if I were to try to explain your points, it might be along these lines:

Entanglement follows rather directly by switching to discussing a bipartite system. I'd talk about a superposition like $|01rangle+|10rangle$. Then do a local measurement of the first bit. What's the result? Well, by the rules QM has for composing subsystems into bigger systems, the result would again be intereminate, but we can know what the other bit's state is! And that's the essence of entanglement.

Uncertainty Principle

What if we try to measure two properties of the system, not just one? Say, we try to measure both the speed and position of a particle?

Since every measurement in the "strange" cases is probabilistic, there is a certain uncertainty in the result. We might for example get a certain velocity on average, but with other velocities also probable. So we get more-or-less one result, but with an uncertainty.

This depends on whether the state is "strange" for this property, for velocity. The Uncertainty Principle is the fact that there are pairs of properties that are "incommensurate" in the sense that one cannot measure both with perfect accuracy. If the state is not "strange" for one, so that it's value is fixed, then the other will have essentially infinite uncertainty. For example if the state is $|vrangle$ meaning that we'll measure one particular velocity $v$, this means the particle can have any position with equal probability. Which is clearly insane, we don't see such states around us - a particle is in the detector, not on the moon.

So what we end up with usually is a state that's slightly-uncertain for one property, and slightly-uncertain for the other property as well. QM says there is even a link between these uncertainties, which is the mathematical form of the uncertainty principle. But the important point is that a real state has some uncertainty in its location and an some uncertainty in its position as well. We can measure one with great accuracy, but that leads to greater uncertainty in the other.

Interference

Interference can be explained using the two-slit experiment set-up. A particle could have come through slit $|1rangle$ or $|2rangle$. Then on the screen we have a superposition of the two. A "strange" state like $|1rangle+|2rangle$.

Now when we measure on the screen, we don't measure whether the particle arrived from slit 1 or 2. We just measure whether the particle is there. And that's an entirely different property. So it turns out that when we add up the factors for writing the state for this property, they can add or substract from each other. So you end up with states like $|thererangle-|thererangle=0$, i.e. a state that gives "does not exist" at some point on the screen! The state of coming from slit 2 cancels out the state of coming from slit 1, so that (for the property "is the particle hitting the screen there?") the probability becomes zero!

So what you end up with is a pattern of stripes of probability on the screen, areas of high probability and areas of low probability. Not because the particle moved though one slit or the other, but because it moved in a strange state, a superposition, of doing both.

This is not because of the property we're measuring, "being there", but rather it's a general result of QM. The factors can add up or subtract from each other (technically it's a bit more complex (hah hah) than that), so you get this result called "interference".

Quantization

Finally, what gives the theory its name, quantization. I don't really know to explain this from the principle, I think it's due to boundary conditions. So perhaps something as follows.

I'd start with the idea that location measurement can have any of a continuum of values. The particle can be anywhere. Similarly a particle can have any energy, and so on.

Often in QM, however, the state is physically restricted. For example the states of an electron around an atom have to be stable, since the electron moves so fast that any unstable-state would quickly become unstable and destroy itself. So if we impose the requirement of stable (tehnicaly called "stationary") states only, we'll find a strange result - that only some states are allowed. The other states just don't compute, there are no solutions to the equations of QM under them.

So for example we find that the electron around a nucleus can have only particular energies. There are simply no stationary states with different energies. There is a state with a particular energy $E_0$ (-13 eV), a state with $E_0/2$, and so on - but no state with energy $E_0/1.4$, for example. There simply isn't such a stationary state.

So when we measure the energy of the state, we always get one of these discrete values. That's quantization. (The state may still be in a superposition, so we can get a spread of results from these possibilities.)

This applies to nearly all quantities in nearly all cases. So we very, very often get "quantization" of the energy, of things like how fast an object spins about its axis, and so on. Quantum mechanics is called "quantum" because of this quantization.