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.

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.

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

- Schrödinger's equation — what is it?
- Schrödinger's equation — in action
- Schrödinger's equation — what does it mean?

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 www.newton.ac.uk to find out more.*

## Comments

## Entanglement

Will somebody please tell me how how particles, separated by millions of light years, can instantly influence each other? Thank you...

## Communication

It might be our concept of space. We cannot picture electrons as particles bumping against each other being a solid cube. I do not mean a plasma 3-D printer cube. Space is made up of cosmic particles that relay instantly the stimulus as passing through an Fe cube.

We live in a ‘sea’ of neutrinos and don’t even know it. Could this be our Conscious media?