How does Quantum Mechanics work?

I’ve spent a few years trying to piece together quantum mechanics, and this article is the result of my work. More than anything, I’ve desired to understand and convey the ideas of quantum mechanics. The equations, the terminology – those are only interesting to me so far as they tell about the inner lives of atoms and electrons and photons. Basically, I want to be able to explain this stuff with my hands.

Unfortunately, this is a taller order for quantum mechanics than any other area of science. In quantum mechanics, we have equations that work extremely well. But we aren’t sure what they mean. We aren’t sure, for instance, if what we do mathematically corresponds directly to what happens in reality – or if the math simply arrives at the same result, even though reality is doing something very different under the hood.

In any other area of science, you could just look a little closer and determine what’s really going on. But quantum mechanics is the description of what happens when you try to “look a little closer” at photons and atoms and such.

And the results are so wild, they’d make for bad fiction.

So if we want to explain quantum mechanics with our hands, we have to make an allowance here – nature may not look exactly like what I’m about to describe! But it certainly arrives at the same results. You can ponder the implications shortly. Physicists have; no consensus has emerged.

One other side-note before we begin: I’m avoiding the standard terminology of quantum mechanics until the final section. I feel that using someone else’s jargon is a way to import someone else’s thinking – which has its uses, but quantum mechanical jargon is so laden with odd connotations and historical baggage that, in my opinion, it’s a hindrance to the beginner. So, to get you up-to-speed with the rest of the world, this piece ends with a hefty glossary tying everything you’ll learn to the “official” terminology. But before that, look out for metaphors and visual explanations that favor clarity over precision.

(Physicists, you’ve been warned 😉)

OK, enough preamble. Let’s dive right in.

As far as I can tell, basically all of the weirdness of QM is due to two simple facts. Neither is intuitive in everyday life, but they’re the foundation of the quantum:

1. Probabilities can cancel
2. Particles can split into many “phantom copies” of themselves

Let’s talk about each of those.

Probabilities can cancel

This probably sounds like an arcane mathematical fact, but the implications are wild. Bear with me.

In quantum mechanics (QM), we do use normal (or “classical”) probability sometimes – but the majority of the time, we’re thinking in a sort of special quantum probability, which has a unique property: probabilities can cancel.

“So you’re saying something can have a -12% chance of happening?”

Kind of! That’s a very useful first approximation.

“OK. But what does it mean for something to have a -12% chance of happening?”

For one, it means it cancels out something that has a (positive) 12% chance of happening.

“That’s ridiculous. A 25% chance of drawing a spade cancels a -25% chance of drawing a club!?”

Your intuition is correct – that’s not quite how it works. For these special quantum probabilities to cancel, they have to be probabilities of the same event happening. Because when we have two ways for the same event to happen, we add the probabilities of the ways.

Take, for example, dice. Say you roll two dice and count the total dots. How many ways are there to roll a 3? There are two ways:

1. The first die is 1 and the second die is 2
2. The first die is 2 and the second die is 1

So the probability of rolling a 3 is the probability of (A) plus the probability of (B).

But now imagine probability (A) is the negative of probability (B). That means the two probabilities will add to zero. And that would mean… you would never roll a 3!

“That’s weird.”

Yes.

Also, if you roll each die individually, they’ll both come up with 1’s and 2’s normally. 12% and -12% look the same when you’re not required to add them together. But if you need to add the probabilities together – as you do when one event can happen in multiple ways – then some strange stuff happens.

“Like rolling a pair of dice all day long and never rolling a 3?”

Exactly! And, good news: that’s like half of the weirdness of quantum mechanics right there. It’s about what doesn’t happen.

In fact, here’s a hack for understanding QM: if you ever hear about a weird quantum result, simply rephrase it in terms of what doesn’t happen, and then rephrase what doesn’t happen in terms of the multiple ways that it could happen adding to 0.

The double-slit experiment – which is perhaps the most famous quantum experiment of all time (and we’ll talk about it in-depth shortly) – fits this mold perfectly. You may’ve seen it before, but in case you haven’t, it’s where you shoot particles (e.g. photons, electrons, etc) one at a time through 1 or 2 very small, very close together slits. The particles that make it through the slit(s) hit a screen on the far side. When you add a second slit, the pattern they make becomes unexpectedly more complex.

This unexpectedly complicated pattern is usually phrased in terms of what happens: “a crazy pattern!”

But let’s rephrase this in terms of what doesn’t happen.

“Photons don’t land in certain areas?”

Yes. And now let’s rephrase that in terms of the multiple ways the non-event could’ve happen. What are the multiple ways that a photon would reach one of those dark bands?

“Uh, either going through the left slit or going through the right slit?”

Yes. Which means the probability of it going through the left slit vs. going through the right cancel out.

“Uhh… Didn’t you say the photons are shot one at a time though?”

Yes.

“How could a single photon travel both paths and cancel itself out!?”

That’s a great lead-in.

Multiple phantom copies

The second weird yet generative insight about QM is that things (photons, atoms, etc) seem to have the ability to split into many “phantom copies” of themselves².

I say “phantom copies” because they’re never observed directly.

“So why talk about them at all?”

Because they do leave a mysterious – and important – trace of their existence. I’ll explain in a minute. But first, I want to state, as succinctly as possible, how these phantom copies work:

Whenever particles lose contact with the outside world, they split into phantom copies that trace out every possible thing the particles could do. However, upon looking, you find each particle in only one state. Only one phantom copy is reality; the rest disappear.

Weird, huh?

Let’s look at a simple example: an atom emitting a photon.

It’s possible to put an atom in a state where it’s all but certain to emit a photon within the next fraction of a second. But if the atom and photon are isolated enough from the outside world – say, in the center of a very large vacuum chamber – then there’s some amount of time during which we could not, even in principle, say where the photon is. During that time – even if it’s only a millionth of a second – the photon splits into phantom copies that trace out every possible path the photon could take. Specifically:

1. We don’t know the direction the photon leaves the atom, so we need phantom copies going all possible directions
2. We don’t know when the photon leaves, so we need phantom copies leaving at every possible time

And, while this will complicate things, and we will return to it later, it’s worth mentioning now:

3. We also need phantom copies of the atom, one for each possibility of when it jumps down an energy level (which it does when the photon is emitted)

Note: I’m only displaying SOME of the photon phantom copies. In reality, they’d cover the surface of the illustration!

You’ll notice the bulk of these phantom photons leave early (rather than later). The specifics aren’t as important as the idea that some outcomes are more likely than others.

For now, let’s clarify these rules of how phantom copies work.

“Lose contact with the outside world”

Quantum mechanics says that if we aren’t currently detecting the state of some particle – and couldn’t, even in theory, determine its state – then, however brief that duration of isolation is, we need to think of phantom copies doing every possible thing the particle could do.

This is a bit of a mindset shift.

First, it forces you to admit you don’t, in a general sense, know the state of a particle. Maybe you know…

1. A photon was emitted from an atom around time x
2. The photon later hit a sensor at location y

But between those two data points, there’s no way to “just keep your eye on the photon”.

If it were a bird, sure. Enough photons bounce off the bird and hit your eyeballs that you can continuously see the bird.

But a photon? Other photons just pass right through it!³ To reliably get a second photon to interact with the first, the second would need so high an energy, the first would veer far off course. The better you know where it just was, the worse you know where it’s going!

So in between these brief, specific moments of knowledge about a particle’s state, quantum mechanics pushes you towards saying the path of the particle is not even a well-defined concept. It’s all mysterious phantom copies, doing every possible thing the particle could do!

“Every possible thing the particle could do”

This framing raises a question. What are all the things that these phantom copies do?

At first approximation, they do every physically possible thing that the particle could do.

But it’s slightly more complicated than that.

Basically, for any property that you measure – position, speed, mass, spin, etc. – if that property could be found to be different values (e.g. a speed of 1 m/s vs 2 m/s), then you need to consider phantom copies for every possible value.

But… if that property is an instrinsic property of the particle (e.g. the mass of an electron), and won’t ever be found as different values, then you DON’T need to consider phantom copies for that property.

That being said, there are phantom copies doing things that are never physically observed – e.g. particles hitting the blank spots on the back wall of a double-slit experiment, or photons moving faster or slower than the speed of light. (I’ll explain more on this soon)

It might be easier to describe just by seeing a bunch of examples:

All this being said, if there’s some property that’s 100% known, and has a 0% chance of changing, you don’t need to consider phantom copies of different values. They only matter to the degree that different outcomes are, at least in theory, possible.

So if a rotating molecule has zero probability of anything bumping it, you don’t need to worry about phantom copies with other rotational speeds. Only once it’s potentially jostled do you have to start thinking of phantom copies with different rotational speeds.

And what jostles the molecule? Phantom copies of other particles, of course! The whole thing is multiplicative.

A single particle – e.g. an electron – traveling through space is easy to visualize.

Like the last such illustration, this shows only a few phantom copies. You’re more likely to see the particle where the copies are denser⁷.

You’re most likely to see it at a certain point, with the probability decreasing as you spread out.

But as soon as you add a second particle, it’s far tougher to visualize.

You need to consider not just every phantom copy from two particles, but every combination of phantom copies. This becomes too much to visualize cleanly.

So, I’ll just note this for now, and we’ll return to it in the third experiment: when multiple particles are involved, it’s easier to visualize entire phantom timelines rather than simply phantom copies.

As a minor spoiler alert, this is why it’s so hard to simulate quantum systems on computers. For every new particle you add, the possibilities you need to keep track of multiply. Today’s supercomputers can only handle a small molecule on a good day.

Fortunately, there’s a saving grace. Despite the ungodly multiplication of possible scenarios, as soon as the system makes contact with the outside world – that is, you – you find that only one scenario actually happened.

Let’s talk about that.

“You find each particle in only one state”

Whenever you look at a particle (and by “look”, I mean “interact with in some way in order to determine its state”), you only ever see it in one state. All those other phantom copies are gone forever. Which state will you see? As best we know, it’s totally random. From all the phantom copies, it’s as if God just picks one out of a hat⁸. But even if we don’t know which state we’ll see, we can figure out the probability of seeing a particular state.⁹

This, by the way, is how cancellable probabilities and phantom copies come together.

Each phantom copy has a probability of being the one that we see. But it’s not a normal probability, with a value between 0 and 1. It’s a special quantum probability, which can sometimes cancel.

And when does it cancel? When two phantom copies end up in the exact same state, but with equal and opposite probabilities¹⁰.

Here is the analog with classical probability. Read down each column.

Before we go on, I want to take a moment and say: this is kind of it. We will dive into how this process looks in a few specific scenarios, but as outlined above, this is the underlying strange process of QM. We have only unsatisfactory guesses as to what such an odd process means about how the universe works. But the rest of this article is unpacking this core idea, rather than introducing new similarly large ideas.

Anyhow, this should be a “scales falling off your eyes” moment with respect to a few things I mentioned earlier:

I say “phantom copies” because they’re never observed directly.

“So why talk about them at all?”

Because they do leave a mysterious – and important – trace of their existence.

The trace of their existence is when they cancel out and certain events don’t happen.

Or:

That being said, there are phantom copies doing things that are never physically observed – e.g. particles hitting the blank spots on the back wall of a double-slit experiment, or photons moving faster or slower than the speed of light. (I’ll explain more on this soon)

The events never observed are times when two or more ways in which those events could occur have special quantum probabilities that add up to zero.

So: the weirdness of QM is in what doesn’t happen, and that’s because of probabilities that cancel.

Before we look at specific experiments, I want to hammer home one point that’s perhaps the most common misunderstanding of QM when scientists explain it to a lay audience. Don’t worry; it’s half review 😉

“Multiple phantom copies” and “I don’t know” are different

So, to review: any time a photon, electron, atom, molecule, set of molecules, etc. could do any number of possible things, you can think of multiple “phantom copies” doing all of those possible things. However, if you could, even in theory, know what actually happened, you’ll only ever find one thing happened. The phantom versions disappear with hardly a trace.

You might be thinking this sounds like the world’s fanciest way of saying “You don’t know what happened. Then you found out”.

But it is not! It 100%, absolutely is not.

There are 2 reasons why “multiple phantom copies” and “I don’t know” are different:

1. Phantom copies that end up in the same state can cancel each other out (If you merely don’t know something, there’s no cancelling involved)
2. As long as the phantom copies remain isolated enough so one outcome is not distinguishable, you can (carefully) modify the probability of various outcomes (not always, but sometimes)¹²

The first of these is the easiest to understand, and if it’s the only thing you get out of this article, that’s totally fine. The second reason is more subtle, and I’ll introduce it via analogy below.

Now, if you’re still with me, let’s talk about 3 classic QM experiments.

The goal here is to build intuition for what the constituent particles of reality spend all of their time doing. Ideally, you want these results to feel not surprising. Accordingly, you may need to re-read these sections a couple of times. But, once you’re there, congrats – you truly grasp the basics of how the smallest building blocks of the universe work!

Let’s get started 😎

Experiment 1: the double-slit experiment

The double-slit experiment is the most famous quantum mechanics experiment of all time. And with good reason! It’s a striking result, impossible to explain without quantum mechanics – yet fairly straightforward to understand with it.

Recall that the central mystery is how the addition of a second slit creates such a complex pattern on the backstop, even as you shoot one particle at a time towards the slits.

Our hack for understanding QM results is to reframe them not as something that does happen, but instead as something that mysteriously doesn’t happen. Then we look for 2+ ways that it could’ve happened, and we find how the special quantum probabilities of the different ways sum to 0.

OK, so, in the double-slit experiment, here’s how that breaks down:

“HOLD UP. Did you just say the probability… oscillates?”

Yeahhh. About that.

How quantum probabilities work

When I said probabilities cancel in QM, it’s not as simple as “0.5” cancels with “-0.5”. That’s only part of the story. Truthfully, these special quantum probabilities aren’t a number, they’re two numbers. They’re – and I almost hate to say this – two-dimensional probabilities.¹³ Yes, it’s as weird as it sounds. No, we don’t know what it means. Sorry.

For classical probabilities, we just think of them as a number – e.g. “0.1” or “0.482”.

But for these special quantum probabilities, it’s easiest to think of them as an arrow (i.e. a vector) – an arrow that starts at the origin and extends to some point on or within a circle of radius 1. Technically, we could achieve the same ends with x and y coordinates, but the arrow helps us think in terms that are more convenient for our purposes: length and angle.

Let’s chat briefly about each.

First, length. This one’s easy. The length of our little arrow of quantum probability is related to the actual probability of the event.¹⁴ The longer the arrow, the greater the probability of the event it represents. This is true no matter what direction the arrow is pointing.

Second, angle. The angle of a quantum probability is something that has no analogue in classical probability – in fact, by itself, it doesn’t have a direct physical meaning at all! It simply changes how multiple quantum probabilities add together.

When do you add arrows of quantum probability together? Just like classical probability, you add probabilities together when you want to find the total chance of something that can happen in multiple ways.

To add arrows, simply connect them tip-to-tail. Depending on how aligned the directions are, two e.g. equal size arrows can do anything from double in length to cancel out entirely!

Also, it’s worth reiterating here: each arrow represents a probability. Every single one for the rest of this article. They’re all just quantum probabilities. Strange and multidimensional, not just a number from 0 to 1. But, in many ways, these quantum probabilities work just like the classical probabilities we’re used to. So, every time you see an arrow in the rest of this article, you can double-check your understanding by asking “what quantum probability does this arrow represent?”

One other notable feature of these quantum probabilities – at least for our purposes today – is that they have a tendency to spin around the origin. Sometimes, they spin over time. Sometimes, they spin as you look at different points in space. Sometimes both!

In fact, the spinning is so important that my visualizations will sometimes ignore arrow length entirely – for instance, when I use a color on the color wheel to represent an angle. Ignoring length is fine for our purposes, since the relevant arrows getting added often have the same length anyways, but physicists keep careful track of all of it.

When the two arrows have the same length, the important dynamic to remember is equal-and-opposite cancels. Here are two ways I’ll show that:

Now you have some background on these special quantum probabilities. Let’s look at how this works in the context of the double-slit experiment.

Quantum probabilities over space

Let’s build things up from the simplest possible example – a single particle (like an electron or photon), zipping through space.

If you were to look at a series of points along a line emanating from the particle’s starting position, you’d find that the special quantum probability of finding the particle at various points along the line spins, so to speak.

It’s worth being super clear on what is doing the oscillating here. It’s not a property of the particle that spins. Rather, it’s a probability that spins (classical probabilities can only grow or shrink – but quantum probabilities, being 2-D, can spin). Specifically, it’s the quantum probability of finding a photon at a particular point, measured for a series of points along a straight line. And yes, this paragraph is highly unintuitive. Unfortunately, it’s pretty fundamental to how QM works, so re-read as necessary.

Now I should say that there’s no direct way to measure the special quantum probability of a particle appearing at a certain point. Actually, if we’re being precise, we can’t exactly even measure classical probability directly. You have to run a bunch of trials and hope the law of large numbers¹⁵ is working today.

But the spinning of a quantum probability isn’t even picked up by repeated trials per se. Remember: the classical probability is tied to the length of the arrow – which isn’t meaningfully changing.¹⁶ The direction of the arrow – which is changing – only comes into play when arrows are added together.

So to see any quantum weirdness at all, you need two identical phantom copies to end up in the exact same place, so their arrows can be added together. And to see complete cancellation (i.e. something that mysteriously doesn’t happen), the phantom copies’ arrows of quantum probability need to be equal and opposite.

Fortunately, the distance it takes for the quantum probability to rotate 360° is simply the wavelength of the photon used in the experiment (or, for electrons or atoms, it’s not as neat – but in all cases, it’s related to the particle’s momentum).¹⁷

That means that, to get two phantom copies to cancel each other out – as they do when their arrow representations are equal and opposite – one has to travel exactly one half of a wavelength farther than the other (or 1.5 wavelengths farther, etc.), and they both have to end in the same location.

Let’s see how that looks.

Quantum probabilities & the double-slit experiment

First, I should say that one does not require phantom copies to travel in straight lines. They could loop and swoop every which way! But as far as this experiment is concerned, every zigzagging path does cancel out, due to a happy accident/“accident” of the math.

That leaves only paths composed of straight lines that we need to worry about.

So let’s just look at what happens on the back sensor.

If the two paths are the same length – e.g. for the very middle spot on the sensor – then the photons arrive with their arrows pointed in the same direction. Those arrows add to an even bigger arrow – and therefore, the highest likelihood of the photon arriving at that location relative to others.

For a spot on the screen just to the right of the center, the left path is slightly longer, and the right path is slightly shorter. This means the quantum probability arrow of the left path will be a little more advanced, and the right path won’t advance as much. They’ll be slightly out of sync, and only add to a moderately larger arrow – representing a moderate probability of the photon landing at that spot.

Finally, if you take a spot even farther to the right, the difference in left/right path lengths will increase further. When the path difference is exactly half a wavelength, the quantum probability arrows for the two paths will be pointed in opposite directions, and they will add to zero. These are the dark bands at which the photon never lands!

Continue on to the right, and the pattern repeats!

Here’s an interactive widget to play around with the variables of the double-slit experiment to get a feel for how things work.

Or, if we simply look at how long the resultant arrow is at every single point, we can visualize the entire display at once.

If you noticed the similarity to ripples on a pool, your thinking is in line with many scientists before you. You can recreate a similar pattern with water.

geralt, via Needpix (public domain)

There is a deep similarity in the underlying math, though let’s be precise. It’s not that the particle itself is a wave. Rather, the quantum probability is wave-like, allowing for even (the phantom copies of) a single particle to recreate the same patterns we see in the waves created by trillions of particles.

The last thing I want to draw your attention to is how profoundly strange this all is. It’s very difficult to describe how this double-slit experiment might work without some notion of the particle exploring different paths and those paths interacting with each other. Again, maybe this isn’t how reality operates. Maybe it only arrives at the same results. But as we’re about to see, there’ll be much more for that theory to explain.

Because experimentally, this is all very real. Double-slit experiments (and related multi-slit setups) have been performed with everything from photons and electrons, all the way to beastly 2,000-atom oligoporphyrin molecules. The specifics vary, but we see the same style of pattern again and again.

Tonomura et al (1989)

Juffmann et al (2014)

Anyhow, if you’re with me this far, congrats! You now have a very tangible understanding of how the most famous experiment in all of quantum mechanics works.

Next, we’ll see how these “phantom copies” don’t just take different paths in space, but can take on different states that even a perfectly stationary particle could be found in.

Let’s look at an example of that, by way of analogy.

Interlude: the parable of the coin

Uncertainty in QM isn’t like a coin toss, whose result you don’t know in midair. Rather, it’s like a loaded coin whose loadedness is changeable until revealed.

Imagine for a second that I give you a very special coin. It does something weird when you follow certain steps:

1. Turn it heads up
2. Flip it, catching it on your wrist
3. Without looking at what it landed on, immediately flip it again
4. Now uncover it
5. It’s tails, 100% of the time

If I showed you this, you’d probably think I was using sleight of hand. But say I gave you the coin, you followed the same steps, and you got the same result – tails every time. What do you do next?

Make some easy money on bar bets!?

Nah, sorry, we’re scientists here. You inspect the coin, find it’s normal, and then see what happens if you look after just one flip!

And when you do many trials of flip-once-then-look, you find that the result is 50% heads, 50% tails. Just like a normal coin.

So what’s up with two-flips-then-look being 100% tails? You try flipping twice before looking again, just for kicks. And it’s 100% tails again. Fine, so that’s still weird.

You guess the obvious next thing to do is flip it three times before looking. You do this many times, and find it is once again, 50% H, 50% T. Hmmm.

Now you’re curious… What about four flips before looking? What will happen then? You check, and indeed, it’s heads every time.

So here’s what we’ve got so far:

“OK, stupid coin, I’ll bite,” you think to yourself, as you embark on a series of 5-flips-then-look and 6-flips-then-look trials. Lo and behold, you find:

And let’s say you go as far as necessary to convince yourself that indeed, this pattern loops endlessly. Despite the existential vertigo you have from handling a magic artifact, you feel like some progress is being made. There are essentially four “states” the coin can be in. And it just cycles through them:

This is actually pretty curious. You notice that your map says “50-50” twice. Now all coins are 50-50 when you flip them once. But this coin has two distinct scenarios where it’s 50-50. And they’re subtly different. Not because of what you observe directly (they both yield half heads, half tails), but because of where they lead to. In one 50-50 state, if you don’t look and flip again, you’ll definitely get HEADS. And in the other 50-50 state, if you don’t look and flip again, you’ll definitely get TAILS. So the coin – or something – must have some memory, some way to track, some deeper variable for where it’s at in this cycle.

Hm.

Then I make one more off-hand comment to you: “You should try waiting 2.41 seconds after the first flip”.

Just given how your day’s going so far, you decide to take my advice. You start the coin at heads, flip it in the air, keep it covered for 2.41 seconds (you’re very accurate with timing), and then look.

Heads. And then tails. And then heads again… After enough trials, you convince yourself it’s 50-50. You’re about to say, “Hey, that doesn’t change anything – it’s still 50-50”, when you realize that this coin has two 50-50 states. If you wait after flipping it once… is it in the same 50-50, or the other one?

So you decide to flip it once, wait 2.41 seconds, then flip it again, and THEN look. And guess what – it’s heads every time!

So you tentatively draw a new path in your diagram:

And then you wonder: if you go from left 50-50 (“L”) to the right 50-50 (“R”), can you do the reverse?

Your most reasonable guess is that by going to R, then waiting 2.41 seconds, you’ll end up at L.

So you try it. You turn the coin to heads, flip it, catch it, wait 4.82s this time, flip it again – and then look. Tails. You do it again, and get tails again. And again. Etc.

Your new theory: if you go L and wait 2.41 seconds, you’ll end up at R. But if you’re at R and wait 2.41 seconds, you’ll end up at L.

You once again sketch another arrow in your burgeoning magic coin diagram:

I appear again and ask if you’ve considered winning some bar bets with coin tricks. You say yes. I scoff and say you should use it to break RSA encryption and make billions. “Though you’d need to be able to read the flip without looking at it…” I mumble.

“Huh?” you say. But you ignore the comment. This day has been weird enough already.

Experiment 2: the Ramsey experiment

In the quantum world, a lot of things act like the coin in the example above – atoms, for instance. I’ll explain how in a second.

But first, I want to talk about why I used the analogy of the coin. There are 2 main reasons:

1. Because it makes it more visceral how strange it is that small systems really act this way (and if you really feel in your bones the weirdness of the coin, it’ll help you remember the weirdness of QM)
2. Because it illustrates the second major reason that “many phantom copies” is different from “I don’t know”

What’s that reason? Well, it’s what I started the parable of the coin with:

Uncertainty in QM isn’t like a coin toss, whose result you don’t know in midair. Rather, it’s like a loaded coin whose loadedness is changeable until revealed.

Do you understand how the loadedness of this coin is changeable as long as I haven’t looked yet? By some combination of (a) flipping it again and (b) waiting, you can change the coin’s results distribution however you’d like.

When you flip a normal coin on your wrist and cover up the result, you don’t know what it is.

But with the “magic” coin, it wasn’t the case that you merely didn’t know whether it was heads or tails. Upon flipping-but-not-looking, the coin actually entered a whole new state – something not captured by “heads or tails”.

In any case, many quantum systems work very similarly to the coin. A simple one is the electron energy state of an atom.

What’s an electron energy state? Well, the electrons around the nucleus of an atom have multiple separate energy levels they can be found in. This is actually kind of counterintuitive. You’d naively expect they could have a range of energies. But not so! They’re either at specific energy level A, or B, or C, or whatever. For our purposes, higher energy levels tend to mean the electron is, on average, found farther from the nucleus.

Today, we’ll only deal with two consecutive energy states – which I’ll call “ground” and “excited”.

New Scientist

These electron energy states map over to the coin analogy from the previous section. But instead of flipping the coin to transition between states, we beam the atom with a perfectly calibrated blast of photons.

A version of the Ramsey experiment using a rubidium atom won the Nobel prize in 2012¹⁸. So… rubidium it is!

So here’s what our “map” looks like for this rubidium atom:

The “ground” and “excited” electron states replace “heads” and “tails”, microwave pulses take the place of coin flips, and the waiting time between 50-50 states is much, much shorter (this picosecond duration can be arbitrarily lengthened in the lab, but for the sake of simplicity, we’ll continue referencing this one stupidly short value). Also, I’ve also renamed the 50-50 states to “Plus” and “Minus”.

For our little rubidium atom, if you pulse it with a precise blast of microwave radiation, and then look at its energy state, it’s ground 50% of the time, excited 50% of the time.

Again, if you did not know quantum mechanics, you might think, “Ah, we’ve found the blast of radiation that makes it 50-50 ground or excited, and we simply don’t know which it is until we look!”

But if, instead of looking, you keep it isolated, wait precisely 9.8 picoseconds, and then blast it again, you’ll see that it’s 100% ground state.

Alternatively, if you wait twice as long between blasts – a whole 19.6 picoseconds – the subsequent second blast will reveal an excited state atom every single time.

On one hand, you’re probably like “Yeah, that’s exactly how the magic coin works, NBD”. But on the other hand, it means the magic coin is actually real (!)

Blast the atom once, it’s 50-50. But if you don’t look, and instead blast it again – with very precise timing – you can choose whether you want it to appear 100% ground or 100% excited.²⁰ No coin on earth works that way.

Now the core idea of this article is that quantum weirdness is because of (1) cancelling probabilities and (2) phantom copies. So let’s write out this experiment in that framework:

The last two steps are new – we haven’t covered those yet.

But rather than try to explain this over paragraphs of text, here is what is looks like:

One confusing thing about this visualization is why the excited phantom copy, when blasted, forms a ground copy with its arrow flipped. I’ve researched what this means, and I’ve come back empty-handed. If I find out a satisfying answer, I will update this article²².

And here is the whole sequence in one picture, if you’d rather see it laid out end to end:

Imagine not knowing about QM. What would you think the blast was doing?

“Either exciting the atom or letting it stay in the ground state – like, maybe the photons miss? – each possibility with 50% probability”

Right. You’d think the atom had a definite state, excited or ground – you simply didn’t know which.

But how can you prove that this situation is different? How can you prove this is something different than mere ignorance of the atom’s state?

“Because if you do another blast, you can get 100% ground or 100% excited, depending on how long you wait between blasts!”

Exactly. And it turns out that phantom copies with cancellable probabilities are what we need to use to visualize how this works.

In the double-slit experiment, there were potentially infinite phantom copies with identical properties taking different paths.

In the Ramsey setup, there are two phantom copies with different properties located at the same point in space.

Next, let’s take the final conceptual step for this article. Let’s examine what happens when there are two particles that each split into phantom copies.

Experiment 3: the Hong-Ou-Mandel experiment

The Hong-Ou-Mandel experiment is perhaps the simplest experiment that forces us to reckon with how QM deals with multiple particles. Since our goal is to be able to visualize things as tangibly as possible, I’ll keep the lede front and center. From earlier:

when multiple particles are involved, it’s easier to visualize entire phantom timelines rather than simply phantom copies

This will make more sense shortly.

In the Hong-Ou-Mandel experiment, we have two photon sources, two sensors, and a beam splitter.

What’s a beam splitter? It’s a half-mirror. When a photon hits it, one of two things happen, each with equal probability:

• The photon passes through (like glass)
• The photon reflects (like a mirror)

Here’s what that looks like. You can click the photon sources to get a feel for how things work.

Maybe you’re wondering, “what happens if I fire both photon sources at the same time?” Well, that is the experiment!

First, try predicting how that would go.

(Yeah, I know. This is a quantum mechanics article. You probably think there’s going to be something tricky – and you’re right! But we haven’t covered enough for you to know how the trickiness will go, so just work it out as if QM didn’t exist)

A few moments of thought will convince you that there are 4 scenarios with equal likelihood:

1. The left photon passes through AND the right photon passes through
2. The left photon passes through AND the right photon bounces
3. The left photon bounces AND the right photon passes through
4. The left photon bounces AND the right photon bounces

Because (1) and (4) both lead to both sensors going off, that means the breakdown of results would be:

• 50% of the time, both sensors will go off once
• 25% of the time, the left sensor will go off twice
• 25% of the time, the right sensor will go off twice

But that is, of course, not what happens.

In order to see what happens, I need to give you two additional pieces of information. Here’s the first: when a phantom copy of a photon reflects off a beam-splitter, its special quantum probability rotates by 90°.²³

If we’re visualizing the angle with color, then you’ll notice the reflected copy has a color that’s 90° further ahead than the copy that passes through.

What does this mean? Again, for classical probability, we can only compare probabilities by whether they’re larger or smaller. “Probability x is 25% smaller than probability y”. But for quantum probabilities, we have a new method of comparison: how out of sync they are, as measured in degrees.²⁴ “Quantum probability x is 90° out of phase with y”. The out-of-syncness doesn’t affect the classical probability of seeing the particle there, only how that phantom copy cancels with other identical copies at that location.

Now, contrary to how we worked through the double-slit experiment, we’re not going to predict the Hong-Ou-Mandel experiment by drawing rainbow lines. Why not? Well, remember what these rainbow lines are: they’re the quantum probability of seeing the particle at that point, as measured at every point along the path. They’re the probability of a specific event happening.

In our case, since there are two photons, we’re actually interested in two events happening. Sensor A detecting a photon (or two) AND sensor B detecting a photon (or two). And there’s no clean way to show all the possibilities with little rainbow lines towards the sensors.

But we can use the same spirit. Rather than visualizing the quantum probability of a single event at different points in 3-D space, let’s find the quantum probability of events in event space.

Put simply, we’ll take the 3 end results we’re interested in…

1. Sensor L goes off twice
2. Sensor R goes off twice
3. Sensor L and sensor R each go off once

…and we’ll figure out a quantum probability of each occurring.

There is one way for sensor L to go off twice (Photon L reflects AND photon R passes through).

There is one way for sensor R to go off twice (Photon L passes through AND photon R reflects).

But there are two ways for R and L to each go off once:

1. Photon L passes through AND photon R passes through
2. Photon L reflects AND photon R reflects

Now, to figure out which end result happens, we simply need to find the quantum probability of each one.

The first and last scenarios are slightly easier. The middle scenario – in which each sensor goes off once – involves adding two possible ways that could happen. But let’s start with the easy part.

So, take the first scenario, in which the left sensor goes off twice. There are two things that need to happen for the left sensor to go off twice: one photon passes through AND the other photon reflects. To figure out the quantum probability of such a scenario, we need to multiply (a) the special quantum probability of a phantom copy passing through with (b) that of it reflecting.

Multiplying quantum probabilities

“What’s this? We’re not only adding quantum probabilities, now we’re multiplying them?”

Yes – but it’s pretty straightforward.

Think of classical probability. Whenever we want to know the probability of A AND B happening, we multiply the probabilities of A and B together.²⁵ For instance, the probability of a person being born male in May is the probability of being male (~0.5) multiplied by the probability of being born in May (~0.083).

Well, our special type of quantum probability is no different than classical probability here.

“But how do you ‘multiply’ two arrows together?”

Math actually provides a very straightforward answer here: you add the angles and multiply the lengths.²⁶

In the context of the Hong-Ou-Mandel experiment, two things make this even easier:

1. Since all paths are the same length, all possible arrow lengths (at the sensor) are the same, and we can basically ignore the exact value²⁷
2. The exact direction of the arrows doesn’t matter – only their relative directions compared to each other. So we can treat those pretty loosely too

So we’re doing arrow multiplication on easy mode 😉

Quantum probabilities in the Hong-Ou-Mandel experiment

Given the above, we can say that the quantum probability of the left sensor going off twice looks like this:

It’s a slightly shorter arrow in a different direction. Ok, fine.

The same is true of the right sensor going off twice:

Where things differ is when each sensor goes off once. In that case, we have two possible timelines that each involve two concurrent events. So we will do two arrow multiplications, then add those results together.

First, when both phantom copies pass through the beamsplitter:

And finally, when both phantom copies reflect at the beamsplitter:

Now, because both of those results are indistinguishable, we need to add their quantum probabilities. And when we sum them, we get the most devious of quantum results…

Zero! The paths cancel. The timeline in which both photons reflect has a quantum probability equal and opposite to the timeline in which both pass through.

Therefore, it never happens that each sensor goes off once.

No matter how many times you shoot both photon sources simultaneously, both photons go right or both photons go left. They never split.

It’s a bit crazy, no?

And yet it’s reality.

Jachura and Chrapkiewicz (2015)

In these experiments of quantum mechanics, a theme starts to emerge. The stuff the universe is composed of, when no one’s looking, endlessly splits into copies of itself in every permutation of possible paths and properties. The whole system is accounted for in an odd type of probability, a probability that spins (trillions of times per second, even). It’s a probability that cancels too, and while anything that can happen one way might occur, some things that can occur many ways are simply never seen.

With that ends my main exposition of QM. However, one task remains. I have rigorously avoided all QM terminology and jargon up to this point. If you are ever to read about or speak with someone about QM, they will have no idea what you’re talking about with “phantom copies” and “cancelling probabilities”.

So let’s translate the very tangible, physical picture thus far into the language the rest of the world uses for the concepts and ideas above.

How to discuss QM at a cocktail party

A half-dozen pieces of QM jargon probably set my understanding of this stuff back by years. Many of the words we’re stuck with raise more questions than answers. Hence my description above, which focused on tangible explanations and metaphors.

But now, it’s time to learn how everyone else talks about this stuff. If you’ve heard some of the terms below before, I hope you have a renewed appreciation for the strange physicality of this science.

Let’s begin.

Amplitude

I’ve talked a lot about “special quantum probabilities” in this article, but I should correct myself. In QM, there is a sort of thing like a probability that can cancel, but it’s called an “amplitude” (or a “probability amplitude”). Mathematically, not only can it be negative (whoa!), but it’s actually a complex number (double whoa!).

(The spinning arrows of length ≤1 we’ve been using? Those are all amplitudes)

Whenever you hear the term “amplitude”, you can always just think “special quantum probability that can cancel” and you’ll be totally fine. The two ideas are totally equivalent.

You use the term “amplitude” the same way you’d use the word “probability” too. Instead of saying “there’s a probability that the photon will do X”, you’d say “there’s an amplitude that the photon will do X”.

For what it’s worth, to get the classical probability of some result, you do the following:

1. Take the probability amplitude for each separate way the result could occur
2. Add them together
3. Take the length of the result (the magnitude of the amplitude 🧐)
4. Square it²⁸

Why square it? No one knows, but it probably says something very deep about reality. Let me know if you find out!²⁹ 🤷‍♂️

Superposition

Whenever I’ve talked about “multiple phantom copies” of a particle or system, that’s what physicists call a superposition. If you’ve read the least bit of QM, you probably figured that one out 😉

The biggest misconception about superposition is that it kinda sounds like a fancy way of saying “I don’t know what state the thing is in”. But remember: that’s wrong for two reasons:

1. Amplitudes in a superposition can cancel³⁰, meaning a certain event simply may never be observed (these weird cancellations aren’t explainable by “I don’t know what state the thing is in”)
2. You can tweak a superposition (in ways you can’t tweak “I don’t know what state the thing is in”), which we saw in the parable of the coin and Ramsey experiment

Any property of a particle or a group of particles that can vary can be in superposition. In general, spatial superpositions get the most airtime (“the particle is everywhere!” 👻), except for in quantum computing, where we really carefully put things in a superposition of just two states (like the rubidium atom in both its ground and excited energy state).

But as long as a particle or system is isolated from the outside environment, it can enter a superposition of every possible state it could be in. And when you sum up all those possibilities, well, we have a word for that…

Wavefunction

The mathematical expression of the probability amplitude of every possible state of the system is called the wavefunction.

(It’s kind of like a probability distribution, but with amplitudes instead of classical probabilities)

So if you e.g. close your eyes and throw a ball, you could make a probability distribution of seeing it in different places when you open your eyes. At every point in space, you’d have a certain probability of seeing the ball there.

In QM, if you e.g. close your eyes and throw an electron (I’m only half-joking), you’d have to work out the calculations in amplitudes, not probabilities (because of weird cancellations, remember?). But at every point in space, you’d have a certain amplitude of seeing the electron there. And that mathematical expression is the wavefunction!

So, next question: why “wave”?

OK, remember how amplitudes tend to oscillate? (i.e. the arrows rotate?) For the double-slit experiment, we saw them oscillate over different points in space. In the Ramsey setup, they oscillated over time. Mathematically, something oscillating over space and time already is a wave. You just need to look at the full picture to see it.

Sadly, this organic technicolor sloshing is going to come to a screeching halt. The multitude of possibilities represented with the wavefunction will be replaced with a single reality – as soon as it becomes possible to tell the state of the particle. However, the jargon for this is a bit odd…

Measurement/Observation

Ohhh boy. This one’s a doozy.

So a superposition is a very delicate thing. If a particle or system is in a superposition, and it subsequently becomes possible to tell – even in theory – what the actual state of the particle or system is, then the superposition immediately disappears and the thing is only in one state. All those phantom copies vanish without a trace.³¹

An air molecule in Earth’s atmosphere might make it all of a nanosecond before it slams into another air molecule, and the multitude of tiny paths its wavefunction describes over that nanosecond get collapsed into a single position.

Nonetheless, historically, we were mostly concerned about superposition collapsing in the context of laboratory experiments. Unfortunately, we’ve inherited, as the terms of art for what collapses a wavefunction, measurement or observation.

This means if you hear someone say “a photon measured the state of the system”, they’re not claiming the photon is conscious, nor are they saying the lil’ corpuscle of light is performing science. Instead, they’re saying a photon interacted with the thing that was in superposition such that it’s theoretically possible to glean some information about the system’s state. And the wavefunction has now collapsed.

This unfortunate jargon has also led some scientists to say things that sound far more mystical than were probably intended – e.g. “The very presence of an observer changes the results of the experiment”, etc.

But that’s only the second-most famous problem with quantum measurement…

The Measurement Problem

So this whole bit about “when you look, all the phantom copies – except for one – disappear” probably feels a little weird. Why just one? You’re saying nature just throws out a huge number of things it just did?

When a system is isolated from the broader environment, its wavefunction (which, remember, gives an amplitude for every possible state the system could be found in) evolves over time in a very mathematically clean way. It’s deterministic, it’s reversible. Mathematicians love it.

However, when measurement happens, it’s the opposite. We go from some number of possible states – two, twenty, a million, infinity – to just one. And even worse, that one appears to be chosen at random. There’s no way to mathematically “roll back the clock”. And of course, because it’s random, you can’t predict it in advance either.

Scientists have felt this tension very acutely for a century, and it’s called the measurement problem.

This may only hit with the math folks, but it’s basically akin to saying:

(A + B) × C = AC + BC = AC

Why? Because I threw out the AB, that’s why!

Having some sense of the measurement problem will contextualize the next few terms in our glossary.

Interpretation

Most physics equations map to reality in a pretty straightforward way.

For instance, the equation for the motion of a ball thrown through the air involves all the things you’d intuitively think of that could affect this – the angle and speed of the throw, how much air resistance there is, the pull of gravity, etc.

However, QM is different.

With quantum mechanics, we discovered equations that predicted reality stunningly well – we just don’t know what they mean.

And so QM has something that no other part of physics has. It has interpretations. Again, these different interpretations all have the same equations and the same predictions.³² They’re simply about what those equations mean.

In some sense, an interpretation is an answer to the measurement problem.

Let’s look at 2 of the most famous.

Copenhagen interpretation

This is the default, “textbook” interpretation of QM. Unfortunately, it doesn’t actually do that much interpreting – it sort of just throws its hands up in the air and says “That’s the way it is!”

For instance, if you ask a Copenhagen devotee what it means for an electron to be in a superposition, they’re likely to say the question is meaningless (“you can only ask about the result of a measurement”) or even say the math is the real thing (“the electron is the wavefunction, nothing more”).

Does this resolve the measurement problem? Not really. It’s simply labelled the “collapse of the wavefunction” – and not really explained in any further detail.

The steelman of this is that the mathematical axioms of QM reference “observing”, but they don’t actually specify what an “observation” is. So Copenhagen devotees are merely refusing to indulge in speculation beyond what the math requires.

Well, some Copenhagen devotees refuse to indulge in such speculation. It’s worth noting that others took the implications of “observation causes the wavefunction to collapse” and just ran with it.

Observation by what? By conscious entities, of course! This is a nice way to tie in almost any metaphysical system you’d like into otherwise materialist physics.³³ If true, this has the neat implication that reality is sort of lazily rendered, like a video game that only loads the part of the map nearest you.

(Einstein disliked this, famously asking, “Do you really believe the moon is not there when you are not looking at it?”)³⁴

So it’s worth noting that, if you meet someone who subscribes to the Copenhagen interpretation, they may be (a) a quantum physicist who cares more about predicting reality than explaining reality, or (b) Deepak Chopra.

Many Worlds Interpretation

“Many worlds” is the understatement of the century. It should properly be called something like “infinite parallel universes spawning every femtosecond”.

According to the Many Worlds Interpretation (henceforth “MWI”), there’s not a universe in which quantum events happen randomly (or, as Einstein put it, “God does not play dice”), but a deterministic multiverse, in which every possible event that can happen does happen – just in different branches.

It’s pretty similar to the “phantom timeline” idea I’ve mentioned above. The MWI take on the double-slit experiment is that, for anywhere one could see the particle land, there’s a branch in which the particle did land there. And if you shoot 100 particles through the double slit, the universe splits into every possibility, every time.

Of course, it’s not just quantum mechanics experiments that split the multiverse. The implication is that basically every particle interaction does. The roughly 10⁸⁰ atoms of the known visible universe are constantly spawning an unfathomable number of branches with every collision!

If this sounds wild, I’ll grant you that. But in its favor, it is a very clean resolution to the measurement problem. Why does the wavefunction collapse into a single state? It doesn’t! Sure, it appears to, here in our branch. But the rest of the wavefunction is still alive and well, spread out across other branches of the multiverse!

But MWI is not a monolith, and proponents debate even some basic ontological questions about branches:

• Are branches discrete, countable things?
• Are there many branches, or infinite branches?
• Do the new branches truly branch off from existing ones, or did they all always exist?
• If they branch, when exactly do they branch? – when we lose contact with the particle, or regain it?
• Why do the rules of probability for branches have amplitudes associated with them? What’s that all about?

But one thing all MWIers agree on is that there are an effectively infinite number of copies of you. The branch in which you started reading this paragraph will turn into an ungodly number of branches containing slightly different yous – all by the time you finish this paragraph. Then, in even more branches, you will go on to live every life that it is physically possible for you to live.

Putting aside the existential vertigo, MWI is perhaps the easiest interpretation to visualize. It’s no coincidence that “phantom timelines” (and, for single particles, “phantom copies”) are so similar to branches in MWI. Two relevant differences: (1) in my writeup, I’ve more or less ignored the measurement problem (i.e. what happens to the non-observed copies) and (2) I’ve consistently described the phantom copies branching off as soon as the particle loses contact (many modern MWI proponents believe branches aren’t separate until the particle regains contact, a process called “decoherence”).

But if you’re willing to allow branches to separate the moment anything loses contact, then that opens up perhaps the strongest argument for any interpretation of QM I’ve heard, and that’s from David Deutsch, the godfather of quantum computing.

However, in order to understand his argument, we need to cover the basics of quantum computers. Buckle up! 😎

Quantum computers

Quantum computers & quantum computing (henceforth: QC) are easiest to explain at the lowest level and the very highest levels. All the stuff in the middle would require another article-length explanation.

The major low-level idea behind QCs is: they use qubits instead of bits.

A bit can be 0 or 1, and “classical” (i.e. normal) computers do all their computation, file storage, and input/output using bits.

A qubit, on the other hand, can be 0, 1, or a superposition of 0 and 1.

Given that the #1 most common misunderstanding of superposition is it’s just a fancy way of saying “I don’t know”, this sounds a bit like trying to augment your computer with random coin flips. Probably not much alpha there 🤷‍♂️

But you, wizened reader, know that you can perform operations on superpositions. Indeed, the whole Ramsey experiment section was on just this. An atom is not like a regular coin; it’s like a magic coin, with a sort of state that only shows itself in heads and tails, but, under the hood, is more complex.

But even that analogy doesn’t make it obvious how QCs offer any advantage over normal computers. To understand that, let’s look at our diagram of phantom timelines in the Hong-Ou-Mandel experiment:

Now imagine this. Every time you fire off the photons, you create 4 branches in the multiverse, each with its own amplitude. Since the middle two scenarios have (a) indistinguishable end states, and (b) amplitudes that sum to zero, they cancel. Those branches disappear outright.

If you understand that, then you can easily understand the high-level idea behind QC algorithms: you do calculations using a quantum system (photon, atom, etc), then get the branches containing the wrong answer to cancel out, such that only the branch with the correct answer remains. And if that’s the branch that remains, that’s the branch where you and I will find ourselves!

Since the purpose of this section is to prep you for cocktail party chit-chat, I should say that the most common misunderstanding of QC is emblazoned across the header of the internet’s most popular QC blog, Scott Aaronson’s Shtetl Optimized:

If you take nothing else from this blog: quantum computers won’t solve hard problems instantly by just trying all solutions in parallel.

Indeed, while QCs can try all solutions in parallel, there’s no guarantee that you’d find yourself in the branch of the multiverse containing the correct answer. For that, you need much more cleverness – making QC algorithms preeetty complex compared to traditional computer science fare.

Nonetheless, Shor’s algorithm – the most famous QC algorithm – was developed early in the history of QC (1994). It allows for much faster factoring of large numbers than classical computers can achieve, which would notably render many current encryption algorithms useless. Nonetheless, post-quantum cryptographic algorithms do exist, and we’ll undoubtedly move to them before the first quantum hacker steals trillions 🤞

Speaking of Shor’s algorithm, this is also related to David Deutsch’s justification for MWI. He asks, and I’m paraphrasing:

If you were to turn all matter in the entire observable universe into a classical computer that spent billions of years factoring an unimaginably large number, and then you were to do the same calculation almost instantly on a quantum computer that fits on your desk, where did that second computation happen?

Deutsch, of course, believes there’s only one answer: quantum computation happens in the multiverse! It happens in the incalculably many branches that are always being generated – but that we can sometimes carefully harness to cancel in juuuust the right way so that we find ourselves in a branch where the quantum computers are correct.

This is a stunning idea – but he’s far from having convinced the rest of the physics community³⁵, and in general, the field called “Foundations of Quantum Mechanics” – which deals with what is happening under the hood – remains fascinating.

To end our glossary, we’ll switch gears to something much simpler, and more famous.

Schrodinger’s Cat

This is perhaps the most classic thought experiment in all of QM.

Imagine we have a chamber that is completely sealed off from the outside world, such that its contents can enter a superposition. In that chamber, we place:

• A cat
• A robot with a gun pointed at the cat
• A quantum system with two states (such as a rubidium atom in its ground or excited state)
• A measurement device that can determine the quantum system’s state

The robot puts the quantum system in a superposition. Then, the robot measures its state. If the state is e.g. HIGH, the robot shoots the cat. If the state is LOW, it lets the cat live.

Now, since the whole setup is isolated from us, the whole setup is in a superposition. But it’s an absolutely wild superposition, in which there are two possibilities:

• The atom was measured HIGH, the robot shot the cat, and the cat is dead
• The atom was measured LOW, the robot didn’t shoot the cat, and the cat is alive

Superpositions aren’t just for atoms anymore! They’re also for cats, which have phantom copies that are both alive and well and bleeding out on the floor.

Erwin Schrodinger, the OG quantum physicist who discovered the wavefunction, proposed this thought experiment to show just how ridiculous the implications of the theory were. Jokes on him though, nowadays we just kinda go with it. “Yup, cat’s alive and dead. NEXT!”

(Unless you’re a MWI proponent, who says “Yup, cat’s alive in 50% of branches, dead in the other 50%. NEXT!”)

And because I can’t let a good thing go, I’ll mention a third interpretation, called “objective collapse”, which debates whether large macroscopic systems (e.g. a cat) can be in a superposition. Perhaps they naturally break down past a certain size! After all, we’ve definitely put some largish molecules into one, but nothing like a cat. And so, they propose, the cat’s maybe/maybe-not death chamber would collapse into a single state well before any robot assassin was reading off the energy state of an atom in there. NEXT!

And all the rest

QM is not a small field, and trying to do it justice in an intro will always leave stones unturned.

Nonetheless, there are foundational ideas and terms we haven’t covered. But, if I’ve done my job, you’ll pick them up quickly:

• The Mach-Zehnder Experiment. The single best experiment to read about next. It’s the same effect as Ramsey – but with a photon’s path instead of an atom’s energy level.
• Quantum tunneling. Good cocktail party chitchat, but it’s basically just fallout from what we’ve already talked about.
• Heisenberg’s Uncertainty Principle. New material³⁶. A grab-bag of quantum effects that all pertain to the unknowability (or undefined-ness) of pairs of properties. Being concise about this is above my paygrade.
• Entanglement. Easy to understand the basics³⁷, but the in-depth version (e.g. Bell’s Theorem) is fiendishly complex. Nonetheless, entanglement is the zen master version of how the quantum truly differs from the classical.
• Wave-Particle Duality. A piece of jargon, that in Bill Watterson’s words, “makes language a complete impediment to understanding”. Still, you hear this term all the time, and it’s worth knowing what it is – and isn’t.
• The Schrodinger Equation. The math governing how the wavefunction changes over time. Lots of complex³⁸ calculus.
• Basis. Not your typical cocktail party quantum fare, but if you’re thinking “particles are just phantom copies, easy”, this will keep the ontological weirdness flowing.

In a sufficiently broad conversation on QM, these will all come up. However, even if this article can’t cover them, you will find them much simpler to understand given the tangible model of things we’ve talked about over the last 15,000 words.

Oh, and one last thing – what’s the difference between superposition and not knowing the state of the system? 😉

Further reading

• QED by Richard Feynman. A layman’s intro to the quantum lives of photons and electrons (called “quantum electrodynamics”, hence the title). Explains how a wide array of common light effects (reflection, refraction, fluorescence, etc) work at the quantum level. Notably, written by the greatest science communicator of all time, Richard Feynman (who also discovered some of this stuff and won a Nobel prize for it 😉). Absolutely my most recommended read for those who liked this article.
• Quantum Country by Michael Nielsen & Andy Matuschak. A giant 4-part article on QC and QM. Michael Nielsen has (literally) written the textbook on QC, and while he’s one of the world’s best explainers of technical concepts, warning: this piece comes with the full mathematical formalism a practicing physicist would be interested in! Nonetheless, it builds up an understanding of quantum computing – including a quantum search algorithm – from the ground up. Incredibly good.
• LessWrong Quantum Sequences by Eliezer Yudkowsky. For those comfortable with math (imaginary numbers, linear algebra), this is a surprisingly accessible introduction to QM. I recall it as being overly-wordy – and pretty smug regarding how MWI was obviously the correct interpretation of QM. Nonetheless, a worthwhile read for someone diving in.

Thanks to Gautam Shine, Scott Aaronson, Mithuna Yoganathan, Abhi Vyas, Matt Favero, and Steven Young for their feedback. LLMs were consulted in the research of this article, but any hallucinations are my own. I welcome further feedback.