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

In the previous article, I wrote about one of the most famous experiments of all time. Young’s double-slit experiment and its variations impressively reveal many of the bizarre surprises that quantum mechanics has up in its sleeve. As long as we think of our world in terms of classical physics, the quantum phenomena are mostly incomprehensible, although perfectly describable by quantum mechanics.

The double-slit experiment.
(By Koantum, svg version by Trutz Behn (Own work) [CC-BY-SA-3.0], via Wikimedia Commons)

Theoretically, we’ve already fired macroscopic particles (paint droplets) and light (in form of photons) through a double slit. I hope that the difference between the intensity distribution of classical particles and of quantum-mechanical particles (photons, in our case) became clear to you.
In shooting macroscopic particles, the intensities of the rays partially going throught the respective slits are simply added together. The result intuitively is the overall intensity. One could just say: $|\psi (x=D,y)|^2 = |\psi_1|^2+|\psi_2|^2$. (The “$\psi(x,y)$” represents the wavefunction which we use to describe particles quantum-mechanically. Consider it as the probability density of the particle to be found at a certain location. In classical physics, we can just sum up the intensities of the individual waves going through the respective slits and write $I_\text{total}=I_1+I_2$.)
If we, however, use photons instead of paint droplets, we clearly observe an interference pattern behind the double slit, as long as both slits are opened! Thus, the intensity is suddenly angle-dependent, which we can read out of the following equation. It’s the $\cos\dots$ (“cosine”) that causes the dependence on an angle (cf. the previous article):
$|\psi(x=D,y)|^2 = |\psi_1+\psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + (A^2/r^2)\cdot\cos k(r_1-r_2)$

If one of the two slits is shut, the interference phenomena immediately vanish and we observe the classical distribution of intensities again.

The following figures should serve as an illustration of the difference between the classical and the quantum-mechanical outcome of the double-slit experiment.

Intensity distribution of classical particles behind a double-slit.
(By Klaus-Dieter Keller [CC-BY-3.0], via Wikimedia Commons)

Intensity distribution of light. Both slits are opened.
(By Klaus-Dieter Keller [CC-BY-3.0], via Wikimedia Commons)

All that I said up to this point, I already explained in detail in the previous article. Those of you who have the feeling that the development of the above reasoning was too prompt might want to check out my earlier blog post.

Let’s try something new today – by firing electrons against the double slit – and see what happens!

(You can find all the other articles of this series here.)

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Electrons

Well, we need electrons. So we replace out light source with an electron source and our photographic plate behind the double slit with a suitable electron detector.

By the way: Electron sources often are just a hot and bare wire which emits electrons due to the conatined thermal energy. Then the electrons can be accelerated in a desired direction using an electrical potential. Sometimes, not as much technology is involved as one could possibly imagine, right? 😉

Also, we scale the slit width, the space between the slits, and the distance between the double slit and the detector at the ratio of the light’s wavelength $\lambda$ to the de Broglie wavelength $\lambda_D$. Or, to put it simply: We have to alter the spacings of the experiment when using electrons instead of photons in order to obtain beautiful results. (Probably I should note at this point that, in doing so, the physical implications don’t change at all!)
After adjusting the experimental setup, we turn up the electron source aaaaand … Wow! We observe interference phenomena which are completely analogous to the intensity patterns of light – although we are using electrons!

Intensity distribution of electrons behind a double-slit
(By Klaus-Dieter Keller [CC-BY-3.0], via Wikimedia Commons)

Once again, we see strong indications that matter can indeed be described in terms of wavefunctions!

As soon as we send individual electrons through the double slit (the interference doesn’t disappear in doing so!), a pressing question arises: Through which of the two slits did the respective electron actually go? Since it is indivisible, it must go through either one slit or the other, right?
Can we find a way to trick nature and find out which paths are “chosen” by the particles before they reach the detector?
Let’s try!

In order to gather informations about the electron paths, we modify the experimental setup: We place a detector behind the first slit which shall register the electron if it passes through slit number 1. (The following reasoning would also be plausible if we put the detector behind the second slit!) Additionally, the electron shall be able to pass the detector unhinderedly. It’s hard, if not impossible, to actually build this kind of detector, but since we are on a theoretical journey through the world of quantum physics, let’s don’t bother about these restrictions of reality. 😉

Which experimental outcome do we expect? Will there be an interference pattern? Or not?
Well, we still hope to observe interference. Also we hope to get the information about every single atom, if it went through slit #1 or if it didn’t. This will be indicated by the detector responding to an eventual electron pass: If the little light on the detector flashes, we know that the electron was seen at slit #1 – it therefore went through this slit on its way to the big detector located behind both slits. If the little light remains dark, the electron must have gone through slit #2. …At least, that’s what we hope to see in order to shine some light on the mysterious electron behavior.

Weeellll… the experimental results show how wrong you can sometimes be! After switching on the electrons source, the electrons zip through the slits. The detector behind the first slit flashes in half of all cases, but we cannot see interference happen at all. The individual electrons, thus, don’t follow the interfering probabilities which were valid before we put the additional detector behind a slit anymore. However, how does an electron going through slit #2 (no detector is placed there!) “know” if there actually is a detector at the other slit without ever being near the first slit? Imagine the following situation: There are some points in the typical interference pattern on the backmost detector upon which no electron is allowed to impinge. These are the points of destructive interference. But if one slit is closed, the interference pattern disappears and suddenly the electrons are allowed to hit the detector at the points of destructive interference.
Somehow an electron going through the detector-less slit manages to “know” if we placed a detector behind the other slit or not, since it never arrives at the “forbidden points” of destructive interference when both slits are open and no additional detectors are put into the setup. But it does impinge at these very points when the other slit is closed or equipped with a additional detector.
How does an electron know what there is at the other slit without ever being there?

Well, as you see, classical physics really comes under pressure of failing to offer a satisfying explanation.

According to classical physics, interference phenomena are related only to waves, not to particles. But in quantum mechanics, particles are described via wavefunctions. This opens up new possibilities.
(By Lookang many thanks to Fu-Kwun Hwang and author of Easy Java Simulation = Francisco Esquembre (Own work) [CC-BY-SA-3.0], via Wikimedia Commons)

Quantum-mechanically treated, the problem appears to be relatively clear, though:
If we consider the “normal” version of the double-slit experiment (no additional detectors attached), the electron’s wavefunction propagates through both slits simultaneously. But since the wavefunction represents is a probability density function, it just gives the probability of the electron to be found at a certain location. “The higher the wavefunction at a location, the likelier it is to find the electron there.”
As is turns out, the points of impact of individual electrons are indeed random! But although being determined by pure chance, they are equipped with certain probabilities which only come to light if we repeat the experiment lots of times. The random individual points of impact gradually add up and form the known interference pattern. The following pictures should illustrate the just-discussed process. (This figure, too, already appeared in the previous article.)

Electrons are sent through a double-slit. A clear interference pattern emerges over time.
(By user:Belsazar (Provided with kind permission of Dr. Tonomura) [CC-BY-SA-3.0], via Wikimedia Commons)

As soon as we put any kind of detector behind slit #1, everything changes!
However, if we think about the wavefunction spreading out throughout the experimental setup, the observed results seem to be logical.
Okay, let’s activate the electron source once again and see what happens:
The electron’s wavefunction approaches the slits, somehow passes through them (we don’t know yet how this happens exactly), and finally manifests itself at a (random) location on the backmost detector behind the slits. In the previous experiments, we didn’t know anything about the electron’s path. Did it go through slit #1 or slit #2? All we could do was to make a point about probabilities. But this time, there is an additional detector behind the first slit which can be consulted. Has it seen the electron at slit #1?

• Yes, the detector reports an electron pass through the first slit. Hence, we know that the electron definitely went through this slit.
• No, the detector doesn’t make the slightest peep. The electron, therefore, passed through the other slit.

What we observe in both cases, however, is a classical intensity distribution (thus no interference!) – that’s the same result as we got when we alternatingly shut one of the two slits (cf. the previous article):

If we put an extra detector behind one slit, the intensity distribution of electrons is equivalent to the classical distribution.
(By Klaus-Dieter Keller [CC-BY-3.0], via Wikimedia Commons)

The additional detector behind one slit already acts as a measurement, at which the electron wavefunction gets “destroyed”. As soon as we ask for any kind of information about the electron path, we restrict the wavefunction so much that it actually propagates through only one slit. And as a matter of fact, a wave going through just one slit cannot cause interference – no matter if it’s a water wave, probability density wave, etc.!

There’s another interpretation of the observations:
Before the electron appears on the detector behind the double slit, it takes every possible path simultaneously. It’s state, thus, is a superposition of all these paths. The paths can interfere with each other and in doing so they create an interference pattern.
However, if there’s an extra detector at one slit, this superposition is destroyed. As soon as the particle is detected at one slit, it cannot be in a superposition state when it arrives at the backmost detector anymore, since it already is at the respective slit.
If the detector at the slit doesn’t indicate an electron pass, the line of argument works completely analogously: Now the electron did not go through the detector-equipped slit, but through the other one. The superposition of all possible paths is destroyed once again and all we can observe behind the slits is a classical distribution of intensity.

Also, we can illustrate the whole issue mathematically (don’t be afraid, that’s neither very complicated nor necessary for understanding the point of this article!):
$|\psi|^2 = |\psi_1+\psi_2|^2 = |\psi_1|^2+|\psi_2|^2 + \psi_1^\ast\psi_2 + \psi_1\psi_2^\ast$.

The asterisked $\psi$‘s stand for the complex conjugate, which doesn’t have to bother you further. The overall wavefunction $\psi$ therefore consists of the wavefunction $\psi_1$ (which represents the case that the electron went through slit #1) and $\psi_2$ (the electron went through slit #2). If we want to know the probability for finding the electron behind the double slit, all we have to do is to calculate the square of the absolute value $|\dots|^2$. After doing some mathematics, we exactly get the expression above, in which the two terms $\psi_1^\ast\psi_2 + \psi_1\psi_2^\ast$ are responsible for interference.

Now that we think of the electron in terms of its wavefunction, let’s consider the experimental outcome once again:

• The extra detector sees the electron passing through slit #1:
This immediately implies that $\psi_2$ has to be equal to zero, since the electron cannot take the way through the second slit.
As a consequence, $|\psi|^2 = |\psi_1+\psi_2|^2 = |\psi_1|^2+|\psi_2|^2 + \psi_1^\ast\psi_2 + \psi_1\psi_2^\ast$ turns into $|\psi|^2=|\psi_1|^2$, just because of the fact that $\psi_2=0$ (and therefore $\psi_2^\ast=0$). All interference terms disappeared and the intensity distribution looks like in the case of just one open slit.
• The detector does not register an electron pass through the first slit:
Now we know that $\psi_1=0$. Equally, this results in $|\psi|^2 = |\psi_1+\psi_2|^2 = |\psi_1|^2+|\psi_2|^2 + \psi_1^\ast\psi_2 + \psi_1\psi_2^\ast$ transforming into just $|\psi|^2=|\psi_2|^2$. And again we observe the intensity distribution of one slit (only that, this time, it’s slit #2).

Needless to say, there are other subtle variations of the experiment trying to squeeze out informations about the electron/photon/… paths, but they always fail in this respect.
As soon as any kind of information about the exact way of a particle exists, the interference pattern blurs and results in a classical distribution of intensity. Thus we can state: Interference phenomena arise out of a fundamental lack of knowledge of the particle’s exact path through the experimental setup.

If we learn something about the path the particle takes, this is akin to performing a measurement. This measurement destroys the wavefunction. (In the sense of the Copenhagen interpretation of quantum mechanics, physicists call this event the “collaps of the wavefunction”.) It also alters the wavefunction’s future spatial and temporal evoltion. Briefly speaking, every measurement on a quantum system changes its very state – it destroys the previously existing superposition of all possible states.
(Out of this considerations, gedanken experiments like “Schrödinger’s cat” emerge. Although being a substantial contributor to the development of quantum mechanics, Austrian physicist Erwin Schrödinger refused to believe that the laws of quantum mechanics differ that much from our day-to-day experiences. This is why he invented the example of “Schrödinger’s cat” – he wanted to illustrate how preposterous and weird this new branch of physics seemed when compared to the macroscopic world, i.e., classical physics. Schrödinger held the belief that, one day, another interpretation of quantum mechanics with the potential of fixing these “early interpretation difficulties” would be formulated.)

Today, too, I have a conclusive video for you in order to relax after this load of words. It’s from the wonderful YouTube channel MinutePhysics and it’s about (Schrödinger) cats.

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