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A ridiculously short introduction to some very basic quantum mechanics

"I think I can safely say that nobody understands quantum mechanics."
Richard Feynman.

Quantum mechanics was developed in just two years, 1925 and 1926 (see here if you want to know why). There were initially two versions, one formulated by Werner Heisenberg and one by Erwin Schrödinger. The two tuned out to be equivalent. Here we'll focus on the latter.

The general idea

Schrödinger's version of quantum mechanics built on a brain wave of the young French physicist Louis De Broglie. In 1905 Einstein had suggested that light might behave like waves in some situations and like particles in others (see here). De Broglie figured that what goes for light might go for matter too: perhaps tiny building blocks of matter, such as electrons, might also suffer from this wave particle duality. It's a strange concept but don't go thinking about it for too long at this stage. Just carry on reading.

Cosine wave

A snapshot in time of a vibrating string. The wave function describes the shape of this wave.

Ordinary waves, such as those that can travel down a piece of string, can be described mathematically. You can formulate a wave equation, which describes how a particular wave changes over space and time. A solution to that equation is a wave function, which describes the shape of the wave at every point in time.

If De Broglie was correct, then there should be a wave equation for those matter waves too. It was Erwin Schrödinger who came up with one. The equation is of course different from the type of equation that describes ordinary waves. You might ask how Schrödinger came up with this equation. How did he derive it? The famous physicist Richard Feynman considered this question futile: "Where did we get that [equation] from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger." (You can find more mathematical details about Schrödinger's equation here.)

A solution to Schrödinger's equation is called a wave function. It tells you things about the quantum system you are considering. But what things? As an example, imagine a single particle moving around in a closed box. Solving the wave equation which describes this system, you get the corresponding wave function. One thing the wave function doesn't tell you is where exactly the particle will be at each point in time of its journey. Perhaps that's not surprising: since the particle supposedly has wave-like aspects, it won't have the clearly defined trajectory of, say, a billiard ball. So does the function instead describe the shape of a wave along which our particle is spread out like goo? Well, that's not the case either, perhaps also unsurprisingly, since the particle isn't 100% wave-like.

The strange consequences

So what is going on here? Before we continue, let me assure you that Schrödinger's equation is one of the most successful equations in history. Its predictions have been verified many times. This is why people accept its validity despite the strangeness that is to follow. So don't doubt. Just carry on reading.

Erwin Schrödinger

Schrödinger's equation is named after Erwin Schrödinger, 1887-1961.

What the wave function does give you is a number (generally a complex number) for each point x in the box at each point t in time of the particle's journey. In 1926 the physicist Max Born came up with an interpretation of this number: after a slight modification, it gives you the probability of finding the particle at the point x at time t. Why a probability? Because unlike an ordinary billiard ball, which obeys the classical laws of physics, our particle doesn't have a clearly defined trajectory that leads it to a particular point. When we open the box and look, we will find it at one particular point, but there's no way of predicting in advance which one it is. All we have are probabilities. That's the first strange prediction of the theory: the world, at bottom, is not as certain as our everyday experience of billiard balls has us believe.

A second strange prediction follows straight on from the first. If we don't open the box and spot the particle in a particular location, then where is it? The answer is that it's in all the places we could have potentially seen it in at once. This isn't just airy-fairy speculation, but can be seen in the maths of Schrödinger's equation.

Suppose you have found a wave function that is a solution to Schrödinger's equation and describes our particle being in some location in the box. Now there might be another wave function which is also a solution to the same equation, but describes the particle being in another part of the box. And here's the thing: if you add these two different wave functions, the sum is also a solution! So, if the particle being in one place is a solution and the particle being in another place is a solution, then the particle being in the first place and the second place is also a solution. In this sense, the particle can be said to be in several places at once. It's called quantum superposition (and it's the inspiration for Schrödinger's famous thought experiment involving a cat).

Heisenberg's uncertainty principle

As we have seen, it's impossible to predict where our particle in the box is going be when we measure it. The same goes for any other thing you might want to measure about the particle, for example its momentum: all you can do is work out the probability that the momentum takes each of several possible values. To work out from the wave function what those possible values of position and momentum are, you need mathematical objects called operators. There are many different operators, but there's one particular one we need for position and there's one for momentum.

When we have performed the measurement, say of position, the particle is most definitely in a single place. This means that its wave function has changed (collapsed) to a wave function describing a particle that is definitely in one particular place with 100% certainty. This wave function is mathematically related to the position operator: it's what mathematicians call an eigenstate of the position operator. ("Eigen" is German for "own", so an eigenstate is something like an operator's "own" state.) The same goes for momentum. When you have measured momentum, the wave function collapses to an eigenstate of the momentum operator.

If you were to measure momentum and position simultaneously, and get certain answers for both, then the two eigenstates corresponding to position and momentum would have to be the same. It's a mathematical fact, however, that the eigenstates of these two operators never do coincide. Just as 3+2 will never make 27, so don't the mathematical operators corresponding to position and momentum behave in a way that would allow them to have coinciding eigenstates. Therefore, position and momentum can never be measured simultaneously with arbitrary accuracy. (For those familiar with some of the technicalities, the eigenstates cannot be the same because the operators don't commute.)

As we know from experience, superposition disappears when we look at a particle. Nobody has ever directly seen a single particle in several places at once. So why does superposition disappear upon measurement? And how? These are questions nobody knows the answers to. Somehow, measurement causes reality to "snap" into just one of the possible outcomes. Some say that the wave function simply "collapses" by some unknown mechanism. Others suggest that reality splits into different branches at the point of measurement. In each branch an observer sees one of the possible outcomes. The measurement problem is the million dollar question of quantum mechanics. (Find out more in Schrödinger's equation — what does it mean?.)

Another thing that comes straight out of the maths of Schrödinger's equation is Heisenberg's famous uncertainty principle. The principle says that you can never, ever measure both the position and the momentum of a quantum object, like our particle in a box, with arbitrary precision. The more precise you are about the one, the less you can say about the other. This isn't because your measurement tools aren't good enough — it's a fact of nature. To get an idea of how such a puzzling result can pop out of an equation, see the box on the right.

Position and momentum aren't the only observables that can't be measured simultaneously with arbitrary accuracy. Time and energy are another pair: the more precise you are about the time span something happens in the less precise you can be about the energy of that something and vice versa. For this reason, particles can acquire energy from out of nowhere for very brief moments of time, something that's impossible in ordinary life — it's called quantum tunnelling because it allows the particle to "tunnel" through an energy barrier (see here to find out more).

And here's another quantum strangeness arising from the wave function: entanglement. A wave function can also describe a system of many particles. Sometimes it is impossible to decompose the wave function into components that correspond to the individual particles. When that happens, the particles become inextricably linked, even if they move far away from each other. When something happens to one of the entangled particles, a corresponding thing happens to its distant partner, a phenomenon Einstein described as "spooky action at a distance". (You can find out more about entanglement in our interview with John Conway.)

This is just a very brief, and superficial, description of the central equation of quantum mechanics. To find out more, read

Or to learn more about quantum mechanics in general, read John Polkinghorne's brilliant book Quantum theory: A very short introduction.

About this article

Marianne Freiberger is Editor of Plus.

This article now forms part of our coverage of the cutting-edge research done at the Isaac Newton Institute for Mathematical Sciences (INI) in Cambridge. The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit to find out more.

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