Wave-particle duality: the mystery that shaped quantum mechanics

Before the twentieth century, classical physics offered a clear and reassuring picture of the world. Within the framework of Newtonian mechanics, the universe appeared deterministic, consistent, and composed of particles whose properties were well defined at all times. Physical laws contained no ambiguities or dualities, and measurement revealed pre-existing properties. This assumption of objective realism began to fracture with the birth of quantum mechanics.

Fig. 1. Artistic representation of wave–particle duality. Source: Shutterstock

From classical certainty to quantum duality

In classical physics, particles and waves belong to distinctly different physical categories. A particle is characterised as existing at a specific point in space; its interaction with another particle occurs at a specific location, much like collisions between billiard balls. In contrast, a wave is described as spreading across a spatial region, and its interaction with another wave leads to a superposition, with regions where the waves reinforce each other followed by regions where they cancel out. A familiar example includes water waves.

Classical waves transfer energy through a medium by passing it from one molecule to the next. Before the emergence of quantum mechanics, light was also considered a classical wave. Although electromagnetic waves don't require a medium to propagate, they still possess classical wave characteristics such as frequency, wavelength, and amplitude. As such, they superimpose, creating a typical interference pattern with areas of reinforcement and cancellation. These wave-like properties of light were demonstrated by Thomas Young in his two-slit experiment in 1801 (Fig. 3).

Fig.2. The interference pattern produced by light passing through two narrow slits. The bright frbands form where the crests of two waves overlap, reinforcing one another. The dark bands result from the waves cancelling each other out.

The belief that nature respects our categories was shattered in 1905 when Albert Einstein proposed that light can exhibit a particle behaviour. The photoelectric effect demonstrated that light is not a continuous flow of energy, as previously thought, but is made up of discrete energy packets, later named photons. The realization that something as fundamental as light can exhibit wave properties in one experiment and particle behavior in the other shocked the scientific community. This finding challenged the classical views that had guided physics for centuries, paving the way for quantum theory.

Arrival of the Schrödinger equation

The sentiment was growing that wave-particle duality may be a fundamental feature of physical reality at the microscopic scale. If light, previously known as a wave, can exhibit particle behaviour, then particles can possess the wave characteristics. In 1924, Louis de Broglie made the first step toward laying the foundations for quantum mechanics by proposing a formula that connected the fundamental property of waves, their wavelength λ, to the fundamental property of material particles, their mass m.

Fig. 2. The de Broglie wavelength equation, where λ = wavelength of a particle, h = Planck’s constant, p = momentum, m = mass of the particle, and v = velocity.

The crucial link between particles and associated waves was found. Motivated by this idea, Erwin Schrödinger applied the known physics of waves to the physics of matter and formulated the equation that became a cornerstone of the emerging quantum theory (Fig. 2). Its role in quantum mechanics is comparable to that of Newton's formula F = ma in classical mechanics. The wavefunctions (ψ) governed by this equation can provide a complete description of quantum states.

Fig. 3. The time-dependent Schrödinger equation, where i = imaginary unit, ħ = reduced Planck constant, ψ = wavefunction, t = time, Ĥ = Hamiltonian operator (total energy operator),

The classical foundations used by Schrödinger in deriving his equation raised the expectations that quantum waves should possess physical properties comparable to those of electromagnetic waves. In that case, the two-slit experiment with electrons should exhibit an interference pattern similar to the one observed with light in Young's experiment (Fig. 3). The pattern should appear as a smooth transition between bright and dark bands, without any dots indicating particle-like behavior.

Fig. 4. Double-slit interference pattern produced by red light. Image: Jordgette, Wikimedia Commons, CC BY-SA 3.0. Cropped from original.

The two-slit experiment with electrons could not be conducted at the time due to technological constraints. According to the de Broglie equation, an electron has a wavelength much smaller than that of visible light, given the electron's mass and the speed required for the experiment. To produce an interference pattern, the slits and the distance between them must be extremely small, which was beyond the capabilities of the technology of that time.

Max Born and the birth of quantum probability

During that period, Max Born was working on a related problem, analyzing collisions between electrons and atoms. Previously, the two-slit experiment was interpreted from a classical standpoint. However, if we zoom in on the slits, they become a quantum system, represented by the atoms they are made of. As electrons pass through the slits, they interact with the atoms and are deflected. Born wanted to see if there was a connection between the deflection angles and the interference pattern predicted by the Schrödinger equation.

Born found that the angles align perfectly with the predictions of Schrödinger's equation. This led to the conclusion that quantum waves are not physical phenomena, but rather mathematical abstractions. A wavefunction encodes all the information about a particle and evolves alongside the particle through space and time. As such, it can statistically predict a physical outcome when a particle interacts with another particle.

The wavefunction ψ(x), which describes a particle's evolution through space, can predict the probability of that particle hitting the screen at a given point x. If we directed a stream of electrons through two slits, they would create an interference pattern due to the statistical distribution. The bright bands would highlight regions of high probability, while the dark bands would correspond to regions of low probability.

This perspective marked the rise of the probabilistic interpretation of quantum mechanics. While classical mechanics predicts exact results, quantum mechanics can only predict probabilities, demonstrating a radical shift from classical certainty to quantum uncertainty. The Born rule (Fig. 5) cemented this idea by utilising Schrödinger's wavefunction ψ(x) to determine the statistical distribution of measurement outcomes in quantum mechanics, expressed as the probability density P(x).

Fig. 5. According to the Born rule, the probability of finding a particle at position x is given by the square of its wavefunction ψ(x).

Interestingly, Heisenberg presented his uncertainty principle a year later. Although he utilized a different method, his uncertainty principle can be derived from the Born rule. Indeed, if the quantum wavefunctions can only yield probabilities instead of definite results, uncertainties must be inherently embedded in their structure. 

Copenhagen Interpretation

The statistical interpretation of the interference pattern sparked heated debates about the physical nature of quantum waves. Schrödinger never accepted the notion that quantum waves are merely mathematical abstractions. In 1927, when electrons were scattered by a crystal lattice, a diffraction pattern emerged, analogous to that of light. Diffraction patterns are more easily observed than interference patterns because the gaps through which electrons must pass can be wider. Nevertheless, even with the diffraction pattern, electron impacts were registered by the detector as single events.

The alignment of the crystal lattice gaps with the electron's wavelength, as predicted by de Broglie, validated his wave-particle duality hypothesis. However, what are these waves exactly? On the one hand, their wavefunction is essential for predicting interference and difraction patterns. On the other hand, the crystal lattice experiment observed impacts as individual events.

This paradox demanded timely resolution. The Copenhagen interpretation, led by Niels Bohr and Werner Heisenberg, remains the most widely accepted. Born rule dominated the views, shifting the physical description of reality from deterministic to fundamentally probabilistic. Quantum waves are not directly observable, but their wavefunctions accurately describe their evolution and predict measurable outcomes within the realm of probability.

Consensus was that quantum systems are inherently indeterministic, lacking well-defined properties until measurement provides a definite result. Uncertainty is ingrained in the structures of the microscopic world, which is why physicists have a theory that behaves like a wave but yields a single localised outcome in experiment.

When particles arrive one at a time

Several decades later, with further technological advances, the probabilistic interpretation gained clear experimental support. When a beam of electrons was sent through two extremely narrow slits toward a detector screen, the electrons initially appeared as individual dots. Over time, however, these dots gradually accumulated to form an interference pattern (Fig. 6).

Fig. 6. Gradual build-up of a double-slit interference pattern from many individual detection events. Each dot represents a single particle (electron or photon) detected on the screen. Credit: Bach et al. (2013), New Journal of Physics. Via Wikimedia Commons, CC BY 3.0.

Moreover, when the experiment was repeated with a light source that emits one photon at a time, remarkably similar behaviour was observed. Instead of a continuous flow of energy, the detector recorded single-photon events that gradually built up the familiar interference pattern.

If photons and electrons behaved purely as classical particles, they would produce two clusters on the screen corresponding to the two slits through which they passed. If they behaved purely as classical waves, they would produce smooth bands of varying energy density. Instead, what emerges is something different: bands of high and low probability density, exactly as predicted by the Born rule.

This result captures the essence of quantum mechanics. Quantum objects propagate through space according to their wavefunctions, which determine the probability distribution of where they may be detected. Yet when they interact with the detector screen, they always appear as localized particles.

The apparent tension between these two descriptions lies at the heart of quantum theory. A mathematical structure that evolves like a wave ultimately produces individual particle-like events in measurement. More than a century after its discovery, quantum mechanics continues to challenge our intuitions about the nature of reality. In many ways, our exploration of the quantum world is still in its early stages. As experiments probe ever-smaller scales and higher levels of precision, the foundations of this strange yet remarkably successful theory remain an active and fascinating area of scientific inquiry.