Double-slit Experiment

Plane wave representing a particle passing through two slits, resulting in an interference pattern on a screen some distance away from the slits. ^[1].

The double-slit experiment is an experiment in quantum mechanics and optics demonstrating the wave-particle duality of electrons, photons, and other fundamental objects in physics. When streams of particles such as electrons or photons pass through two narrow adjacent slits to hit a detector screen on the other side, they don't form clusters based on whether they passed through one slit or the other. Instead, they interfere: simultaneously passing through both slits, and producing a pattern of interference bands on the screen. This phenomenon occurs even if the particles are fired one at a time, showing that the particles demonstrate some wave behavior by interfering with themselves as if they were a wave passing through both slits.

Niels Bohr proposed the idea of wave-particle duality to explain the results of the double-slit experiment. The idea is that all fundamental particles behave in some ways like waves and in other ways like particles, depending on what properties are being observed. These insights led to the development of quantum mechanics and quantum field theory, the current basis behind the Standard Model of particle physics, which is our most accurate understanding of how particles work.

The original double-slit experiment was performed using light/photons around the turn of the nineteenth century by Thomas Young, so the original experiment is often called Young's double-slit experiment. The idea of using particles other than photons in the experiment did not come until after the ideas of de Broglie and the advent of quantum mechanics, when it was proposed that fundamental particles might also behave as waves with characteristic wavelengths depending on their momenta. The single-electron version of the experiment was in fact not performed until 1974. A more recent version of the experiment successfully demonstrating wave-particle duality used buckminsterfullerene or buckyballs, the $C_{60}$ allotrope of carbon.

To understand why the double-slit experiment is important, it is useful to understand the strong distinctions between wave and particles that make wave-particle duality so intriguing.

Waves describe oscillating values of a physical quantity that obey the wave equation. They are usually described by sums of sine and cosine functions, since any periodic (oscillating) function may be decomposed into a Fourier series. When two waves pass through each other, the resulting wave is the sum of the two original waves. This is called a superposition since the waves are placed ("-position") on top of each other ("super-"). Superposition is one of the most fundamental principles of quantum mechanics. A general quantum system need not be in one state or another but can reside in a superposition of two where there is some probability of measuring the quantum wavefunction in one state or another.

Left: example of superposed waves constructively interfering. Right: superposed waves destructively interfering.^[2]

If one wave is $A(x) = \sin (2x)$ and the other is $B(x) = \sin (2x)$, then they add together to make $A + B = 2 \sin (2x)$. The addition of two waves to form a wave of larger amplitude is in general known as constructive interference since the interference results in a larger wave.

If one wave is $A(x) = \sin (2x)$ and the other is $B(x) = \sin (2x + \pi)$, then they add together to make $A + B = 0$ $\big($since $\sin (2x + \pi) = - \sin (2x)\big).$ This is known as destructive interference in general, when adding two waves results in a wave of smaller amplitude. See the figure above for examples of both constructive and destructive interference.

This wave behavior is quite unlike the behavior of particles. Classically, particles are objects with a single definite position and a single definite momentum. Particles do not make interference patterns with other particles in detectors whether or not they pass through slits. They only interact by colliding elastically, i.e., via electromagnetic forces at short distances. Before the discovery of quantum mechanics, it was assumed that waves and particles were two distinct models for objects, and that any real physical thing could only be described as a particle or as a wave, but not both.

In the more modern version of the double slit experiment using electrons, electrons with the same momentum are shot from an "electron gun" like the ones inside CRT televisions towards a screen with two slits in it. After each electron goes through one of the slits, it is observed hitting a single point on a detecting screen at an apparently random location. As more and more electrons pass through, one at a time, they form an overall pattern of light and dark interference bands. If each electron was truly just a point particle, then there would only be two clusters of observations: one for the electrons passing through the left slit, and one for the right. However, if electrons are made of waves, they interfere with themselves and pass through both slits simultaneously. Indeed, this is what is observed when the double-slit experiment is performed using electrons. It must therefore be true that the electron is interfering with itself since each electron was only sent through one at a time—there were no other electrons to interfere with it!

When the double-slit experiment is performed using electrons instead of photons, the relevant wavelength is the de Broglie wavelength $\lambda:$

$$\lambda = \frac{h}{p},$$

where $h$ is Planck's constant and $p$ is the electron's momentum.

While the de Broglie relation was postulated for massive matter, the equation applies equally well to light. Given light of a certain wavelength, the momentum and energy of that light can be found using de Broglie's formula. This generalizes the naive formula $p = m v$, which can't be applied to light since light has no mass and always moves at a constant velocity of $c$ regardless of wavelength.

The below is reproduced from the Amplitude, Frequency, Wave Number, Phase Shift wiki.

In Young's double-slit experiment, photons corresponding to light of wavelength $\lambda$ are fired at a barrier with two thin slits separated by a distance $d,$ as shown in the diagram below. After passing through the slits, they hit a screen at a distance of $D$ away with $D \gg d,$ and the point of impact is measured. Remarkably, both the experiment and theory of quantum mechanics predict that the number of photons measured at each point along the screen follows a complicated series of peaks and troughs called an interference pattern as below. The photons must exhibit the wave behavior of a relative phase shift somehow to be responsible for this phenomenon. Below, the condition for which maxima of the interference pattern occur on the screen is derived.

Left: actual experimental two-slit interference pattern of photons, exhibiting many small peaks and troughs. Right: schematic diagram of the experiment as described above. ^[3]

Since $D \gg d$, the angle from each of the slits is approximately the same and equal to $\theta$. If $y$ is the vertical displacement to an interference peak from the midpoint between the slits, it is therefore true that

$$D\tan \theta \approx D\sin \theta \approx D\theta = y.$$

Furthermore, there is a path difference $\Delta L$ between the two slits and the interference peak. Light from the lower slit must travel $\Delta L$ further to reach any particular spot on the screen, as in the diagram below:

Light from the lower slit must travel further to reach the screen at any given point above the midpoint, causing the interference pattern.

The condition for constructive interference is that the path difference $\Delta L$ is exactly equal to an integer number of wavelengths. The phase shift of light traveling over an integer $n$ number of wavelengths is exactly $2\pi n$, which is the same as no phase shift and therefore constructive interference. From the above diagram and basic trigonometry, one can write

$$\Delta L = d\sin \theta \approx d\theta = n\lambda.$$

The first equality is always true; the second is the condition for constructive interference.

Now using $\theta = \frac{y}{D}$, one can see that the condition for maxima of the interference pattern, corresponding to constructive interference, is

$$n\lambda = \frac{dy}{D},$$

i.e. the maxima occur at the vertical displacements of

$$y = \frac{n\lambda D}{d}.$$

The analogous experimental setup and mathematical modeling using electrons instead of photons is identical except that the de Broglie wavelength of the electrons $\lambda = \frac{h}{p}$ is used instead of the literal wavelength of light.

1. Lookang, . CC-3.0 Licensing. Retrieved from https://commons.wikimedia.org/w/index.php?curid=17014507
2. Haade, . CC-3.0 Licensing. Retrieved from https://commons.wikimedia.org/w/index.php?curid=10073387
3. Jordgette, . CC-3.0 Licensing. Retrieved from https://commons.wikimedia.org/w/index.php?curid=9529698