# Development of Quantum Physics IX – The Double-Slit Experiment Is Fascinating! (1/2)

So far, we’ve seen that light consists of a great number of tiny, indivisible energy packets which physicists call light quanta or photons. This picture of light is a relatively new one since light was thought of as wave before. Later on, there was a discovery which was presumably every bit as spectacular and sensational: Things we’ve always called normal “particles” suddenly showed characteristics of waves. As it happens, we can describe electrons, which are ordinary “particles” in classical physics, in terms of matter waves. An electron’s wavelength goes under the name of the de Broglie wavelength. The wavy picture of matter is in perfect accordance with all experimental data gathered so far.

A little later, we learned to interpret the matter wave of a “particle” as a probability density function, which means that we can only know the probability of a particle to be found at a particular point in space and time.
Another weird phenomenon results from the observations and theories above: It is impossible to measure properties like location and momentum of a particle with arbitrary accuracy at the same time. The fact that there is a lower limit for a simultaneous determination of certain particle characteristics is embodied in the Heisenberg Uncertainty Principle. This fundamental principle can also explain why atoms are stable. (I already mentioned this in an earlier article about the Bohr atom model.) There is a lowest energy state for the electrons in an atom which guarantees that they don’t fall into the atomic nucleus like we would expect of classical physics due to the attracting Coulomb force between the negatively charged electron and the positively charged protons in the nucleus.

Today I’d like to describe a famous experiment which unites most of the peculiar and curious phenomenons of quantum physics: The double slit experiment.
Countless books can most certainly be written on this topic, nevertheless I’ll try to keep it short and simple. I will just tell you what happens if we fire big particles, small particles, and light particles (by which I actually mean particles of light: photons) against a double slit.

On closer inspection, this experiment is entirely baffling and mind-blowing!

You can find all the other articles of this series here!

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In superposing (coherent) light waves we observe interference phenomena. Classical physics can explain interference only if light is treated as waves. Hence, the double slit experiment was probative for light actually being waves and not particles.

# What is the double slit experiment actually?

Well, this question can be answered quickly: All we do in this experiment is to fire something (in this expample, something stands for beams of light) against a double slit. This is how a double slit looks like, at least in principle:

A double slit. The black areas absorb light – so it can only go through the white slits.
(By Bautsch (Own work) [CC0], via Wikimedia Commons)

The black areas represent absorbing material, whereas the white lines indicate slits in the material through which light can travel in an unimpeded manner.
If we send waves (light waves, water waves, …) through the double slit, areas emerge behind it where wave peaks meet other peaks (or where troughs and troughs come together) in order to interfere constructively and form intensity maxima. Similarly, there are areas where peaks meet troughs – the waves cancel each other out (i.e. interfere destructively) and create intesity minima.
This could look like the patterns in the following graphic:

Diffraction pattern behind a double slit: At some locations the light waves amplify each other, whereas they cancel each other out at locations in between.
(By Bautsch (Own work) [CC0], via Wikimedia Commons)

In what follows, we will send three different kinds of things through the double slit:

• macroscopic particles
• light
• electrons (in an upcoming blog post!)

# Macroscopic Particles

Let’s take an imaginary spray gun and splash some paint against the double slit! (It doesn’t matter which colour we pick – believe it or not ;-). Choose your favourite colour!) Behind the double slit we attach a sheet of paper, on which the paint (I chose red!) sticks.
The following picture shows how the intensity distribution of the paint (“How much paint is on every single point on the paper?”) will look like.

Intensity distribution in case of classical particles.
(By Klaus-Dieter Keller [CC-BY-3.0 (http://creativecommons.org/licenses/by/3.0)%5D, via Wikimedia Commons)

The grey lines represent the intensities originating from the individual slits – the red line is the sum of these individual slit intensities. It is the paint distribution correspondent to the red line which we actually observe if both slits are open.
There is no sign of interference patterns for macroscopic, classical particles!

# Light

Now we replace the paint spray gun with a light source. Additionally, we reduce the width of the individual slits and the distance between them, so that the spatial dimensions are about the same as the wavelength of the incoming light. (In doing so, the outcome of the experiment looks nicer. But the physical implications don’t change at it!)
If we open just one slit and close the other one, we observe an intensity distribution analogous to classical (macroscopic) particles which go through a single slit (= a grey line in the picture above). This holds for both slits.
But as soon as both slits are opened at the same time, the intensity distribution changes fascinatingly:

Intensity distribution in case of light – both slits are open.
(By Klaus-Dieter Keller [CC-BY-3.0 (http://creativecommons.org/licenses/by/3.0)%5D, via Wikimedia Commons)

An interference pattern – characteristically for waves!

But it will get even more fascinating if we dim the light source insomuch that it emits photon after photon – until only one photon is in the experimental setup at any given moment.
Thus, now we send individual light particles through the double slit. Will there be an interference pattern behind the slits again? (It’s still light, isn’t it?) Or will we see its particle-like character? (Particles cannot produce an interference pattern, just due to the fact that such patterns have their origins in the interaction of something coming from one slit with something coming from the other slit.)
In order to answer this question, we successively fire countless photons through the double slit and capture their points of impact on a photographic plate behind the slits:

Individual particles are fired throught the double slit – after many repetitions a clear interference pattern becomes visible. Particle count: 11(a), 200 (b), 6000 (c), 40000 (d), 140000 (e).
(By user:Belsazar (Provided with kind permission of Dr. Tonomura) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)%5D, via Wikimedia Commons)

Hold on! – We have a contradiction here, right?!
If there are just individual photons in the experimental setup at any given time, it’s only possible for the photons to travel through one slit. In order to go through both slits, the photons would have to somehow split – but that’s simply impossible since we already consider the smallest and indivisible energy packets of light. We clearly observe an interference pattern after “collecting” a big number of photons nevertheless. Hence, it cannot be some kind of interaction between the photons causing the interference phenomenon. So what’s the mechanism behind the interference?

Another variation of the double slit experiment provides further observations:
In this experiment we still send individual photons through the double slit. But this time, we alternatingly cover one slit while the other one is left open. So slit #1 is periodically “open”, “shut”, “open”, “shut”, … Conversely, slit #2 is “shut”, “open”, “shut”, “open”, …
As a result, any sign of interference disappears completely and the distribution of the photon’s points of impact behind the slits once again looks like this:

Alternatingly, one slit is open while the other one is shut. The intensity distribution behind the double slit looks like the distribution of classical particles again.
(By Klaus-Dieter Keller [CC-BY-3.0 (http://creativecommons.org/licenses/by/3.0)%5D, via Wikimedia Commons)

The overall intensity is $I_\text{total}=I_1+I_2$, therefore the sum of the “single slit intensities”. Classical, macroscopic particles give the same result.
So we can make the conclusion that – in order to get an interference pattern behind the double slit – it’s essential not to know through which slit the particle is gone. All we can say in the case of interference is that there’s a fifty-fifty chance that the particle passed through either one or the other slit.

These experimental results are utterly confusing if we keep thinking in terms of classical physics (which we normally do in our everyday life)!

However, quantum mechanics provides an entirely unique description of all the observed happenings!
We describe a photon in terms of a wavefunction $\psi$.
This wavefunction basically consists of two wavefunctions again: $\psi_1$ (the photon passed through slit #1) and $\psi_2$ (it passed through slit #2). Hence: $\psi = \psi_1+\psi_2$.
We already know that a particle’s wavefunction only tells us how probable it is to find the particle at a certain location.

We can calculate this probability by determining the square of the absolute value. (This is one of the most characterstic aspects of the Copenhagen interpretation of quantum mechanics. But I won’t dwell on this here, I’m sorry.)

So we’d like to know the distribution of the probabilities of finding a particular photon for all points on the photographic plate behind the double slit (at a distance of $x=D$). For that purpose we just consider the square of the wavefunction’s absolute value:

$|\psi(x=D, y)|^2 = |\psi_1+\psi_2|^2 = |\psi_1|^2+|\psi_2|^2+\psi_1^\ast\psi_2+\psi_1\psi_2^\ast$.

The last two terms (the ones without the “|” characters) are responsible for the occurrence of interference. (The asterisked $\psi$s stand for the complex conjugate of the original wavefunctions – just in case you ask.) You can also convert these terms into $\psi_1^\ast\psi_2+\psi_1\psi_2^\ast = (A^2/r^2)\cdot\cos k(r_1-r_2)$. It’s not that important to understand the details of this expression – but there’s one thing you can easily see: The intensity distribution behind the double slit depends on the angle between the photon’s points of impact and the original direction of propagation (as seen from the location of the double slit). This is indicated by the “$\cos(\dots)$“, i.e., the cosine. However, all I tried to tell you in this paragraph is that there are different ways of saying “interference occurs”.

If we shut one slit – slit #1, for example -, we automatically set $\psi_1=0$, because the wavefunction of something moving through a closed slit is just zero. Thereby, also $\psi_1^\ast$ is equal to zero and no interference occurs since the angle-dependent part $\psi_1^\ast\psi_2+\psi_1\psi_2^\ast$ of the equation vanishes.
What remains is just $|\psi |^2 =|\psi_1+\psi_2|^2 = |\psi_1|^2 + |\psi_2|^2$. This is equivalent to the classical result $I_\text{total}=I_1+I_2$!

As soon as we treat our physical system quantum-mechanically, we realize that it is the wavefunction of the photons – and not the photons themselves! – which propagates through both slits and therefore is able to produce an interference pattern. But the interference pattern resulting from the wavefunction interfering with itself is not what we can directly observe when we do an experiment – it is just representative for the probabilities of finding individual photons in case we’re looking for them. The point of impact for every photon is indeed random! However, if we repeat our experiment sufficiently often by firing photon after photon through the double slit, the interference pattern, which is determined by the wavefunction’s probability density, becomes visible.
More on all that later…

Well, since this blog post got quite long again, I will approach the case when electrons are fired against a double slit in the next part of this series of articles. Then we will see that there are big analogies between the electron and the light case, fascinatingly. The meaning of information in the universe in relation to the double slit experiment, however, will be elucidated once again.

In order to relax the mind after this quantum mechanical excursion, I’d like to refer you to this Veritasium video in which Derek explains Young’s double slit experiment in an entertaining and most vivid way. (And those of you who followed this series of articles to this point possibly know the answer to Derek’s question at the very end of the video!)

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