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**Definition/Summary**The double-slit (or 'two-slit') experiment clearly demonstrates that individual particles exhibit a wave-like behavior, in that an interference pattern can be shown to build up over time, despite the presence of only one particle in the experimental apparatus at any given time.

**Equations**

**Extended explanation****Introduction: the two-slit experiment:**

Many people arrive at this forum trying to understand the famous “two-slit experiment”, in which single particles are shown to exhibit wavelike behaviour. (A five-minute video description of this can be found here. It’s not

*theoretically*perfect, but it describes the

*experimental*results accurately.)

When people try and explain this experiment in terms of concepts taken from everyday life, they quickly run into difficulties. How can something be both a wave and a particle?

A wave is usually thought of as some kind of disturbance, or a vibration, in a “medium”: something like the surface of the sea, or the skin stretched across the head of a drum, made up of a great many points.

A “particle”, as seen in school textbooks (or the video above!), is a small, localised, distinct object, which most people picture like a tiny billiard ball - it occupies one place at one time; it moves if you push it, in whatever direction you pushed it, and in that direction only. Doesn’t it?

The double-slit experiment, of course, shows that it doesn’t!

So how on Earth do physicists combine these very different ideas - of a wave and a particle - into one single “thing”? And how do we reconcile this behaviour with our experience of billiard balls, or of bullets, or of basketballs, as solid, tangible things which move in whatever direction they are pushed? We only ever

*detect*something that looks very like a particle, even though it behaves like a spread-out wave whenever we

*aren’t*trying to detect at it.

**Louis De Broglie:**

A physicist’s first guess - proposed by Louis De Broglie - was that the particle - of exactly the sort we pictured above - was guided through space by a “pilot wave”. This wave spread through space ahead of the particle, and interfered with itself to form the paths along which the particle could travel.

De Broglie actually

*predicted*the wavelike nature of all particles using this idea, for which he received the Nobel Prize in 1929. But his

*picture*, of paths in space, came to be

*rejected*by most physicists because it could not be extended to systems of more than one particle, or atoms more complicated than hydrogen. A complete theory was needed, one that could describe all microscopic phenomena; and when it arrived, physicists were shocked.

**Describing a wave by a function:**

To explain without maths the results at which physicists have arrived is difficult. To understand the key mathematical idea, imagine standing on the shore of a beach watching the incoming tide. When you see a wave approaching, you probably think of it as a “thing” in its own right, coming towards you. But the water molecules that make up that wave probably won’t be the ones washing round your ankles in a few moments. Instead, they will transfer their energy to the water molecules next to them, which will pass it along in turn towards the molecules nearest the shore. As the tide ebbs and flows, it’s always more or less the same water washing up against the shore. The shape of the wave will also change with time; it might gradually swell as it moves through increasingly shallow water, for example. So how do you describe this wave using maths, when it isn’t really a coherent single thing in the first place? The answer is that you describe the surface of the water, over a large area. You mentally chop the surface up into small squares; as small as you can possibly make them. Then, you associate with each square the

*height*of that square above (or below!) “sea level”- a reference point, with particles at sea level having a “height” of zero. (Points below sea level are associated with a negative “height”.) This height will, of course, change with time. So what you need is something that, given a point on the surface of the water and a moment in time, tells you the height of that point relative to our zero-point. This “something” is called a

*function*. A function is a very general idea in maths; anything which takes some “input” and gives you a number can be called a function. In quantum mechanics, physical systems are described by “wave functions”. Written down on paper, they can look quite like the function you might use to describe our water waves above. But the physical meaning behind them is very different.

**Amplitude squared = probability:**

Water waves move energy from place to place. There’s a simple way to work out how much energy is carried by a water wave. You take the maximum height of the wave above your zero-point – called the amplitude of the wave - square it, and multiply it by a particular constant number.

In quantum mechanics also, the square of the amplitude of a wave function has a physical meaning, but a strange one. It tells you not the

*energy*, but the

*probability*of finding a particle in the state described by that function (and the square of the amplitude at a point gives the probability that the particle will be found in a small region of space centred on that point).

**Quantum theory is probabilistic, not deterministic:**

Quantum mechanics is

*fundamentally*indeterminate - there is no way,

*even in principle*, that you can predict where any individual particle will land on the screen.

It’s only when you make a measurement that you “collapse the wave function” – you find a particle in a definite, but

*randomly*determined, place, and it

*only then*behaves as you would expect a particle to behave.

This is where our analogy to water waves breaks down. In a water wave energy was transferred between particles pushing on each other and working against gravity, but there’s no medium for these probabilistic waves. So what are they?

**Wave or particle?**

The honest answer is that we don’t know. (Search Wikipedia for “Interpretations of quantum mechanics” to learn more about what people

*think*is going on when we aren’t looking.) But the important point is that quantum mechanics forces us to abandon our ideas of particles moving along nice trajectories, and this is the key to understanding the “paradox” of the double-slit experiment. The electron does

*not*split into two halves, each of which passes through the slit; how would this give us a wavelike interference pattern? Nor is it the case that the electron

*is*in fact just a wave, or we wouldn’t observe localised particles. Prior to looking for the electron, we describe it by a wave function that is the sum of the functions we would associate with each slit; this is the only sense in which it can be said to “travel through both slits”.

This is duality.

**Constructive and destructive interference**

These functions have different values at different points (like different points on the surface of the sea had different heights), so they don’t add together in a very “neat” way. Sometimes the two functions both have a value greater than zero at a particular point, and add together to give a result that is greater than the value of either of the original functions: this is constructive interference. (Because our probabilities are related to the

*square*of the wave function, adding two negative numbers together has the same effect!)

But when the two functions are added together at a point where one is greater than zero and one is less than zero, they cancel each other out - at least partially, and sometimes totally. So you end up with no chance (literally!) of finding the particle at various points along the screen: this is destructive interference.

This combination of constructive and destructive interference is the origin of the pattern that we see in the experiment, and is why the wavelike character of an electron only manifests itself after large numbers of electrons have passed through the apparatus: each looks like a single electron upon detection, and large numbers are needed to mark out the areas where electrons cannot arrive.

**Macro and micro:**

So why don’t we see quantum effects in the everyday world? After all, if you throw a basketball with your eyes shut, you expect it to behave the same way as it would if you had your eyes open! In part, the answer depends on which interpretation of quantum mechanics you favour; on what you believe is “

*really*happening”

*when you aren’t measuring a system*.

There are, however, two points of

*general*validity that can be made:

(1) Implicit in our description above was the idea that:

when you take a measurement,

you always find something to be in a definite state;

for whenever you you always find something to be in a definite state;

*see*a macroscopic object, you are in a sense

*measuring*its position.

(2) A consequence of the mathematics describing all of these processes is that:

as the number of particles in the system increases,

the probability of obtaining the most likely result increases,

to the point where it is the probability of obtaining the most likely result increases,

*all but certain*that the most likely result will be observed - hence why the chair you’re sitting on stays in one place if you stand up and close your eyes.

**Extremely accurate statistically:**

If you don’t like the idea that the world around you might not be as “well-behaved” as you’ve always believed it to be, consider that there’s lots of examples in classical physics where we’re just as dependent on systems containing lots of particles to behave in a statistically sensible way as we are in quantum mechanics.

Temperature and pressure, for example, are a convenient way of summarising Newton's second law, but are only statistically sensible for large numbers of particles. Even density, and partial pressures of a mixture, are only extremely accurate statistical approximations.

For example, think about a mixture of sand and sugar in a small bowl. It’s possible that, just by stirring, you could separate the sand and the sugar into the two halves of the bowl. But is it likely?

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!