Post: Double-Slit Experiment
The double-slit experiment, a famous experiment first done in 1801, is a massive achievment that allows us to understand theories like quantum superposition and the wave-particle duality. It compares waves with electrons, and how the results are different from things like bullets. It can be done in many ways, either in a box or in the water. There are many experiments that include light and electrons, but the double-slit experiment shows how quantum principles relate to each other.
Experiment With Bullets
The first step is to picture what will happen when we do the experiment with bullets. Picture a gun. Next to the gun, there is a wall with two slits. In front of the wall, there is a backstop to see the outcome. On the backstop there is a movable detector which moves in the $x$ direction. The bullets can get stuck in the detector. What will happen when the gun shoots the bullets? This isn’t a difficult question. On the backstop, we will see two strips of bullets behind the slits.
The other bullets that don’t go in the slits get stuck on the first wall. As in the picture above, if you close the second slit the probability of the bullet going in the first slit is $P_1$. The probability of the bullet going through the second slit when you close the first slit is $P_2$. If you add those probabilities together, you get the probability of the bullet landing in both slits, $P_{12}$. You will see later in this post that the probabilies don’t always add together to get $P_{12}$.
Experiment With Water Waves
If we do the same experiment but with waves such as water waves, we get a different outcome. When the wave gets bigger and soon approaches the slits, it creates two new small waves at the slits. Soon, the waves will colide with each other, and it will create an interference pattern. The curve on the right, $I_{12}$, is the probability of the wave going through both slits. As you can see, the sum of the probabilities of the wave going through hole 1 and hole 2 don’t match $I_{12}$.
This makes sense, because if you look at the picture, the wave is actually going through both slits. The amplitude of the wave, $\hat{h}$, is a complex number. The probability of the wave going through each hole is the amplitude squared. The probability of the wave going through both holes is the sum of the amplitudes squared. To conclude, we can say that \(I_{12} \neq I_{1} + I_{2}\).
Experimemt With Electrons
Now, it time for the main part of this experiment. We get, perhaps, a “realization” when we do the experiment with electrons. The outcome is the same outcome when we do the experiment with water waves. We get an interference pattern. If you look at the image below, you see a symmetrical wave that has parts that are high, and parts that are low. That is called a probability distribution. The places where the wave is higher, there is a higher chance of the electron landing there.
This mind-blowing realization told physicians that electrons act like waves. This also showed that electrons don’t go in straight lines, but rather in a curvy path. Another realization was that electrons don’t go through one slit, but they go through both slits like in the wave experiment. The electron itself goes through one slit, but the electron’s wave goes through both. The sum of the probabilities of the electron going through each hole is not equal to the electron going through both holes. The amplitude of the electron wave, $\phi$, is a complex number, and again, the sum of the probabilities of the electron going through each slit is not equal to the probability of the probability of the electron going through a slit when they are open.
Conclusion - Final Results
This fascinating experiment showed how quantum superposition and the wave-particle duality are both true in this case. The experiment with electrons and the experiment with water waves had the same results, which showed that electrons act like water waves (wave-particle duality), and the electons go through both slits which showed that electrons can be in many states at once (quantum superposition). Unlike in the bullets experiment, the electrons and waves created an interference pattern.
Summary
1) The probability of an event happening in a experiment
is a complex number $\phi$ called the probability
amplitude, squared.
$P$ = probability
$\phi$ = probability amplitude
$P = |\phi|^2$
2) When an experiment can happen in many ways, the probability
amplitude is the sum of the probabilitiy amplitudes of the
experiment happening in each way. There is interference.
$\phi$ = $\phi_{1}$ + $\phi_{2}$
$P = |\phi_{1} + \phi_{2}|^2$
3) If an experiment tells if one or another event happens, the
the probability is the sum of the probabilities of each event.
There is no interference.
$P = P_{1} + P_{2}$