Double-slit experiment ↩

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created: 2021-11-26 18:41:40
modified: 2022-04-20 19:35:42

In this experiment, electrons are shot at a wall with two slits on it. Behind the wall, there is a detector screen, which can tell us where an electron ended up after passing through one of the slits.

Interestingly, on the detector screen, we get an interference pattern on the spatial distribution of incoming electrons, even when they are shot one at a time. This proves that they act like waves and can interfere with themselves. (See wave function.)

Starting state: the electron starts from the source.

End state: The electron is detected in the point of the detector screen.

Event: The starting state and the end state is specified. The event can occur in two mutually exclusive manners: it can either pass through hole 1 or hole 2.

The emergence of the interference pattern

Let's consider the second postulate and let the probability amplitudes of the different routes be waves:

Where is the wavenumber of the electron with momentum and , are the distances of slit 1 and slit 2 from the point . The wavenumbers of the particles are given by the de Broglie wavelength.

Then in case of :

And with this the probability:

Where .
With this modeling, when , then (we find a local maximum on the detector screen) and when , then (the detected intensity is zero on the screen).

Particle-like properties

Let's look at the third postulate. By having detectors near both slits, we can know which one the particle passed through. Let's call them and . On the figure, they detect the light scattered by the electrons.

In this case, the end state of the electron is not only described by the point, but also whether we detect it in or . Let and be the probability amplitudes of the electron passing through slit 1 scattering detectable light into or . Due to the symmetry of the system, the probability amplitudes of the electron passing through slit 2 then scattering light into or are now and .

With these, the probability amplitude of the electron getting to point and detecting the photon in is:

And the probability amplitude of the electron reaching and detecting the photon on is:

Here we used the second postulate and the fact that the probability amplitudes of conditional events get multiplied.

If we manage to tweak the experiment (for example by lowering the wavelength of the light) in a way so that for the electron passing through slit 1, there is a much larger probability for the photon to be detected in instead of and the same for slit 2. we can (almost) be certain of the slit the electron went through.

Then the probability of detecting the photon in :

And the probability of detecting it in :

The complete probability of the electron ending up in point while knowing which slit it passes through is the sum of the two previous probabilities:

Here we used that we considered two independent events, so we have to sum their probabilities, not sum their probability amplitudes like when the events are mutually exclusive:

The properties described above are related to the collapse of the wave function.