Young’s slits (1803)

Consider a light source shining on a screen containing two horizontal narrow slits, placed very closetogether. The light goes through the slits, and is detected on a vertical screen some distance away.Label the slits 1 and 2 (see figure):

Now, if slit 2 is closed, so the light can only go through slit 1, we get an image of slit 1 on thescreen: the intensity of the light on the screen will be peaked, opposite the position of the slit:

Likewise, if we close slit 1, we get an image of slit 2 on the screen, at a slightly di↵erent place.

Young’s slits (1803) cont.If both slits are open we might expect the total intensity on the screen to be the sum of these,symmetrical about the midpoint between the two slits, something like this:

Instead what we actually see is something more like this:

In the middle there is indeed a peak, but either side are dark strips where there is no light at all!Then there is another peak, then another, falling gradually in intensity as we move away from themidpoint. So on the screen we see alternating strips of light and dark, rather like this:

The alternating strips of light and dark are known as an “interference pattern”, or “interferencefringes”.

Young’s slits (1803) cont.The classical explanation for interference is that light is a wave, travelling through space at speed c.At the midpoint of the screen, the distance to both slits is the same, and the waves from slit 1 and 2arrive in phase:

when added together they thus produce a wave with twice the amplitude, and the intensity on thescreen is large. However, as we move away from the midpoint, we move slightly closer to one slit,slightly further away from the other. Eventually the waves will be out of phase: one will be up whenthe other is down. When we add them we get zero, thus no light at all on the screen.For electromagnetic waves, the amplitude is proportional to the electric or magnetic field in thewave. The intensity of the light is given essentially by the square of the amplitude:

I = |A|2.

For two waves, with amplitudes A1 and A2 (such as those from slit 1 or slit 2) the amplitude of thecombined wave is A1 + A2: the intensity is then

I = |A1 + A2|2,

which is not equal to

|A1|2 + |A2|2 = I1 + I2

just as Young observed. So because we see interference on the screen, light must be in some sense awave.

So if light is a wave, how can it be a particle?!

To resolve this paradox, consider the intensity of the light. Since light comes in “lumps” (thephotons), if we reduce the intensity enough, eventually the source will be emitting photons just oneat a time, and we can see what happens to them.

Now, each photon must leave the source, somehow get through the slits, and then arrive at somedefinite spot on the screen. So we will see photons hitting the screen, one by one, in di↵erent places.If we count the number of photons arriving in di↵erent places on the screen, eventually we mustbuild up the intensity distribution described previously: there will be lots around the midpoint, but insome places close by none at all. So the probability P that a photon arrives at a particular place onthe screen must be proportional to the intensity there

P / I = |A1 + A2|2.

For this reason the amplitudes A1 and A2 are sometimes called “probability amplitudes”.

So if light is a wave, how can it be a particle?! cont.

So far so good. Note that we have no choice but to introduce the idea of probability: though theindividual photons arrive at di↵erent places, we have no way of knowing where any particular photonwill turn up.

Now, however, we have another paradox: does a particular photon go through slit 1 or slit 2? If itgoes through slit 1, then P1 / |A1|2, if it goes through slit 2 then P2 / |A2|2. Surely it must gothrough either slit 1 or slit 2, so the total probability must be P = P1 + P2 = |A1|2 + |A2|2? Butthen there would be no interference: this is not what happens.

The resolution of this paradox is that in this experiment, we were only looking at the screen, not atthe slits, and if we never look at the slits, we can’t say whether the photon went through slit 1, slit2, or indeed something more complicated. However if we do look (say by looking for electronsscattered by the photons passing through the slit), then just at the moment we see which slit thephoton goes through, the interference pattern disappears!

So if light is a wave, how can it be a particle?! cont.

This observation is a result of the “uncertainty principle”, first expressed by Heisenberg in 1927: if�x is the uncertainty in the measurement of the position of a particle (like the photon), and �p theuncertainty in the measurement of its momentum, then the product

�x�p > h,

where h is Planck’s constant again. Thus, if we want to determine which slit the photon goesthrough we need to reduce �x to the separation of the slits. But this means we lose precision in themomentum, �p > h/�x , and because p = h/�, �� > �x . But if we cannot fix the wavelengththis precisely, the interference will be lost – which is just what happens.

To summarise: light is both a particle (photon) and, in some sense, a wave. The wave gives,through the square of its amplitude, the probability to find the particle. When there are alternative(but undetermined) possibilities for the way a process can happen, we can add the amplitudes, butnot the probabilities. Only if we can determine experimentally which possibility actually occurred,should we add the probabilities.

These are the essential ideas of “quantum mechanics”: they resulted in a number of Nobel prizes,notably for de Broglie (1929) for waves, Heisenberg (1932) for amplitudes and uncertainty,Schrodinger (1933), for wave mechanics, and Born (1954) for the probalisitic interpretation.

So if light is a wave, how can it be a particle?! cont.

We now ask the following question: what happens in the double slit experiment if we replace thephotons by electrons? Since the electrons are surely particles, they surely won’t interfere? Wrong:with electrons we get interference just like with photons – electrons are wavelike too, their behaviourdetermined by probability amplitudes just like photons. Thomson and Davisson received the Nobelprize in 1937 for showing this experimentally.

Given this, why in classical physics do light and electrons look so di↵erent? The answer is thatphotons are massless, and carry no electric charge, so it’s easy to make lots of them, even at lowenergy (for example in a candle flame). Electrons however are massive, and it thus takes alot ofenergy (mc

2 in fact) to make one. Moreover they carry electric charge �e, so the number ofelectrons is conserved – they are thus easy to count, and it isn’t di�cult to see that they come inlumps. Electron waves have a minimum frequency: since E = h⌫, p = h

� , E 2 = p

2c

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2c

4, forelectrons

⌫2 =c

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m

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4

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m

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