Superposition and Interference

Two lessons back, we introduced the idea that quantum mechanics allows qubits to exhibit both wave-like and particle-like properties. In this lesson, we shall focus on the former and see how wave properties become useful in formulating accurate quantum algorithms. But before that, let’s recap wave theory.

Wave theory

As you recall from your high school or college classes, waves are disturbances or vibrations in a medium that transport energy from one point to another. There are two types of waves, transverse and longitudinal. In transverse waves, the displacement of the medium is perpendicular to the direction of wave motion. But in longitudinal waves, the displacement of the medium is parallel. We’ll limit ourselves to discussing transverse waves as they are relevant to our subject matter.

We typically define waves using their wavelength, speed, and frequency. The speed is obviously the rate at which the wave travels between two points. The frequency is the number of vibrations per unit time. The wavelength is the distance between two consecutive troughs or two consecutive crests. Places of high amplitude are called crests while places of low amplitude are called troughs. We define their amplitude as the greatest displacement from the mean position. The wavelength $\lambda$, speed $c$, and frequency $f$ are related by the following equation:

$$c = f\lambda$$

Looking at the equation above, frequency and wavelength are inversely proportional to each other, meaning that an increase in one causes a decrease in the other, and vice versa. In other words, lower frequencies mean longer wavelengths, and higher frequencies mean shorter wavelengths.

Wave-like particles

Till now, we’ve only told you that quanta and qubits behave both like particles and waves. The notion of particles being particles follows from the definition. But how do these particles also behave like waves?

Young’s Double-Slit Experiment

In 1801, Thomas Young conducted the Double-Slit Experiment in which the English physicist shot a beam of light at an opaque sheet that had two slits in it. Behind this sheet, he placed a screen. Using this apparatus, Young found out that when the width of the slits was significantly greater than the wavelength of light, the results would be as expected: the light cast two shadows on the screen. However, when he narrowed the width of the slits to distances comparable to the wavelength of light, he would see an interference pattern, which is an important characteristic of waves, on the screen.

Wave interference

This interference pattern, as its name suggests, could only be produced if the light diffracted through the slits, interfered, and created a series of crests and troughs (bright and dark spots) on the screen behind it. Interference is a property that is intrinsic to waves. When two or more waves meet at a position, their crests or troughs interfere with each other and merge, either constructively or destructively, creating a resultant wave.

Visually, scientists have given the idea that the photons or electrons moving towards the double slits are in a superposition of particle and wave-like behavior. This behavior continues until this superposition is observed by the screen behind the slits, collapsing the wavefunction and localizing the superposed duality.

Mathematically, we define interference as taking the algebraic sum of the waves’ amplitudes at the meeting point.

To explain further, constructive interference is when multiple crests or troughs add to each other and produce a bigger crest or trough in the resultant wave. Conversely, destructive interference is when multiple troughs add to multiple crests and cancel each other out, leading to a lower or even zero amplitude in the resultant wave. There’s also mixed interference, wherein some of the waves interfere constructively at some points and destructively at others.

If light behaved only like particles, this interference pattern would not exist. Thus, Young showed that light could behave like a wave as well using the Double-Slit experiment. Experiments would later swap this white light that Young used with lasers and electron beams, obtaining the same results, leading to the same conclusion:

• Light and other quanta that were previously thought of as discrete particles have a definite wave-like characteristic to them as well.

In fact, as the French physicist Louis de Broglie later found out, all particles have a wavelength associated with them! And by extension, everything has a wavelength associated with it, which is given by the following equation:

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