A primer on quantum physics
Last updated
Last updated
"I think I can safely say that nobody really understands quantum mechanics." These are the words of Richard Feynman; en passant, I recommend his wonderful book "Surely You're Joking, Mr. Feynman."
To be clear, I don't really understand it either. It's precisely for this reason that I look for the simplest possible path to go towards a better understanding.
Indeed, the quantum revolution shattered the firm foundations of science established by the 19th century. Principles taken as granted, such as the fact that, when we measure something in a system (like the position of an electron), we just look at it and read some pre-existing value. We look at the electron and find it to be here. Of course, that's where it was and we just learned about it. Obvious, right?
Another more recent principle uncovered by Einstein in his theory of special relativity (1905) is that no information can travel faster than light; obvious to us, seen from the 21st century, right?
And yet, these two principles, known as realism and locality, seem to be challenged in quantum physics. Locality? We have to live with this infamous "spooky action at a distance" in which changing something at some location changes something else far, far away. Realism? What is known as the Born rule, according to which the measure of the position of an electron does not just reveal it, but rather picks out a location at random according to some probability distribution and in a way that cannot be predicted.
But how did this happen? Let's start with a bit of history, and in particular with what has a broad consensus in the quantum physics community.
Let's start with a thought experiment, with a star. This is an experiment that I first discovered in "The fabric of reality" by David Deutsch. In fact, it is not just a thought experiment: astrophysicists experience this first hand when trying to photograph the remote universe.
A star is shining. Physically, the light it emits is actually nothing else than an electromagnetic wave that propagates from it, a bit like water waves, but in the void. At the end of the 19th century, physics were well covered by a theory for waves on the one hand (Maxwell) and a theory for matter on the other hand (Newton).
Now, go away from the star. The intensity of the light you see will diminish (quadratically with the distance).
Keep going away, it will keep diminishing.
Once you are very very far away, it will be barely perceptible, but still there.
But at some point, something funny will start happening: it will star blinking. And as you go away even more, blinking less and less. What you just discovered is a photon: an "atom" of light.
Wait, are atoms -- particles -- not matter? Well, that's exactly it. The difference between waves and matter is no longer so clear as it used to be. In other words, the energy carried by the light waves, is quantized because it does not go down and "spread" forever as you walk away from the star: there is a minimal amount. That minimum amount is carried by a single photon, and it is proportional to the frequency of the light it transports (the factor is Plank's constant, ħ).
This is exactly what gave its name to quantum physics: the physics of quanta. A photon is a quantum of light. Quantum physics means that the universe is "pixelized", like our screens, but we barely notice because it's full 4K.
There is even more surprising. It works the other way round, too. You probably noticed with the curtains of your living room that if light goes through small holes close enough, it forms interference patterns; of course, it is a wave, right? So it's like when you throw a few stones on water, it creates circles, and several circles interfere with each other.
Well, if you throw plenty of electrons through a wall with two slits, you should just see two marks on the screen, one for each slit, right?
No. Not if the slits are very close to each other. This is what you see:
Exactly like light.
So, what do we do with all that?
One of the early insights in quantum theory is that a particle is described by a wave function. That wave function provides some sort of probability distribution over where it can be found, and is the state of that system. In our example, something like this:
However, there is something more. If you look closer, there is a phase on each point of that distribution, you can think of it as little wind spinners like we have in our gardens:
Mathematically it means that this distribution has complex values, not real values. Remember the complex numbers? i, which, when squared, gives -1. But it is best to think of each one of these complex values as an amplitude and a phase, rather than real and imaginary parts (which mean nothing).
So the way I like visualize the curve is with a color system where the hue represents this phase, like so:
The probability (more precisely, density of probability) that the particle is a a certain position is then obtained by the square of the amplitude. And bingo: this is how we get the interference patterns when adding two wave functions (the two slits!). Without going into details, it very roughly has to do with this famous remarkable identity we all saw in junior high school: the 2ab part causes the interference.
In some cases, the two wave functions we add do not overlap; then, no interference.
In "mainstream" quantum physics, this wave function is the best description we can have of any system -- even if later on, other equivalent formalisms were introduced with vector spaces and matrices (a wave function is just a vector in a big space).
To summarize, the wave function describes all the places the particle could be, together with additional information on how probable this is (plus the little wind spinners, we'll come back to that).
But how does this wave function evolve, when time elapses? Well, this is given by an equation you probably heard of: Schrodinger's equation -- yes, Erwin Schrodinger, the owner of the most famous cat in the world.
I will not scare you with this equation; instead, let me explain it intuitively: whenever the amplitude is high, it causes the tiny wind spinners (the phase) to rotate. The higher, the faster. But with different amplitudes, the wind spinners don't all rotate at the same speed, so this causes these rainbow patterns in space; and these rainbow patterns then "push" the amplitude curve left or right, cause it to spread, etc.
So you can recognize that the wave function behaves a bit like a wave (with the rotating wind spinners) and like matter (because it "moves" left and right). Schrodinger's equation is kind of a mix between a wave equation (describing water waves, for example) and a diffusion equation (describing how, when you open the window in winter, the cold air enters the room and spreads).
Now comes the hard part, because I will have to describe something mainstream even though this is in disagreement with my (and others') scientific belief (I'll come back to this).
What you may not know (because you heard that quantum physics is random) is that the Schrodinger equation is completely deterministic -- and reversible. So far, so good.
But things become more difficult (understand: controversial) when we want to measure something about a system. Even just looking at a particle already means we are measuring its position (via a photon reaching a detector, for example). Then, something strange happens: observing this particle forces the particle to pick a position, at random, according to the probability distribution specified by the wave function (this is called the Born rule), and the wave function collapses to a wave function squeezed in a narrow band around that chosen position.
This is why quantum physics makes statistical predictions: if, say, the theory says a particle prepared in a given state has a 50% chance of being here and 50% of being there, and you prepared 1,000,000 particles in that state and go ahead and measure, then you will find +/- 500,000 particles on each side (with a reasonable -- understand "typical" -- margin of error). But the theory is not powerful enough to make individual predictions, for each particle.
So, is this theory incomplete? That's what Einstein thought, and he co-wrote a paper with Podolsky and Rosen in 1935 to prove their point.
A core assumption made hitherto, called realism, is that a measurement merely reveals an element of reality that must have pre-existed. There should thus be nothing in the way of a better theory to predict the outcome of any measurement.
The EPR paper applies the realism assumption to the wave function representing a pair of particles that are sent away far from each other (their wave function is defined on ℝ^6 rather than ℝ^3), following by measurements on each of them. The hope is to find a contradiction, but they actually end up discovering a consequence of quantum theory now fondly called the spooky action at a distance.
Concretely, the spooky action at the distance means that if the choice (by the physicist) of which measurement is made on one particle were changed, then the outcome of the measurement of the other particle would change, too.
This is, at least at first sight, in direct contradiction with another well accepted principle: locality, i.e., information cannot travel faster than light, changes cannot propagate faster than light.
So is a local and realist theory, i.e., a theory containing hidden variables where the outcomes of the measurements are already known, somewhere, and with no faster-than-light communication, adequate to describe reality? Several proofs seem to say no.
The first famous result is from Bell, in the year 1964, which came as a direct answer to the EPR paper. Bell showed that any local and realist theory makes predictions in such a way that an inequality on its "output" probability distributions is fulfilled; a very simple one, with just simple arithmetics. This is known widely as the Bell inequality (even though there are actually many such Bell inequalities that "constrain" local and realist theories, discovered later). Problem: this inequality was shown to be broken in quantum experiments (notably, by Alain Aspect in 1982), which seems to exclude a hidden variable approach.
Later came the Kochen-Specker theorem, in 1967. This one shows in a troubling way that Nature is contextual. The proof is purely based on the mathematics of vector spaces. The idea, simply put, is that a measurement consists of (i) a choice of measurement basis (set of vectors) by the experimenter and (ii) a choice of vector in that basis by Nature (the new wave function after the "collapse").
Since experimenters choose their measurement basis freely, a deterministic theory that predicts the outcome of measurements should make such a prediction for any choice of basis. Problem: the Kochen-Specker theorem says that, for systems complex enough, there cannot exist such a mapping from bases to vectors that respects basic consistency rules.
Later, Conway and Kochen, in 2006, published their free-will theorem, which can be very elegantly summarized with this quote:
If indeed there exist any experimenters with a modicum of free will, then elementary particles must have their own share of this valuable commodity.
And yet. There is a theory initiated by Louis de Broglie in 1927 and rediscovered by David Bohm in 1952, which is fully deterministic and predicts the outcome of every measurement. It is known as the pilot-wave theory, as the De-Broglie-Bohm theory, or as Bohmian mechanics.
And it makes the same predictions as quantum theory.
And all quantum physicists agree that it does, i.e., this is not controversial -- just exotic. A curiosity worth mentioning as an anecdote in a footnote.
And, even better: it is remarkably beautiful.
The idea is that particles have a well-defined trajectory (yes!), and that the wave function guides this trajectory. How? The N particle trajectories, represented as a very big vector in ℝ^3N (the configuration space), follow the phase gradient (in this big vector space) of the wave function (which attaches a complex number to each point in ℝ^3N). Are you familiar with machine learning? if so, you might recognize a form of gradient descent! But it goes on and on and on forever, because the phase is periodic.
Is it such a big surprise? We already saw something like this in the Schrodinger equation: the spatial gradient pushes the "amplitude" (wave packet) left or right. Well, the De-Broglie-Bohm theory says it literally pushes the particles.
So if we now zoom out, if we know the initial conditions (the initial wave function and the initial position of each particle), then we are able to calculate all the trajectories, deterministically.
What happened is that we pushed all the probabilities to the beginning of the universe: everything random is just in the initial conditions, which we don't know. But if these initial conditions are distributed according to the wave function (the squared amplitude of it), which is known as quantum equilibrium, then it is proven that the trajectories remain distributed according to the wave function, i.e., the quantum equilibrium is preserved.
When there is a measurement, the wave function splits into two (practically) non-overlapping parts, like so. This is called decoherence.
The distribution of initial conditions induces the probabilities of each measurement outcome, and the probabilities for predicting measurement outcomes are thus the same as predicted by the Born rule; but the difference with the interpretation as a "collapse" of the wave function is that there is this trajectory, and it is known in advance.
After each decoherence, the newly created partitions never interact with each other ever again, so that, zooming out, the overall pattern of the evolution of the universe looks like so:
Each non-overlapping partition corresponds to a "possible world" at any possible time. Each trajectory, at every decoherence moment, goes through one of them.
But wait... we just said there are impossibility theorems that say this can't be! The key here is to understand that the De-Broglie-Bohm theory does not assume that humans are unpredictable, unlike impossibility theorems. And this was well known to Bell in 1964. In fact, he knew about the De-Broglie-Bohm theory before his famous inequality, and this theory even motivated him to do so.
But thanks to Louis de Broglie and David Bohm, we have this alternate way of looking at nature: particles jointly surfing down the phase of the wave function just like boats navigate on the see.
Thanks to them, quantum theory suddenly makes a whole lot of sense.
But let us instead throw particles through a wall with two slits and let them hit a screen on the other side. Let us do it with electrons. This is known as the double-slit experiment. The picture is taken from .
So, it should now be clear that part of the reason why we find quantum physics so complicated and unintuitive is that we assume , i.e., the unpredictability of the choice of measurement basis by the experimenter. As a consequence, our interpretations become limited and must involve probabilities, measurements that change the system, spooky action at a distance, etc. It is so deeply anchored in our beliefs that it is hard to let it go.
