Why Light Behaves Both Like a Wave and a Particle: From Newton to Schrödinger

In classical mechanics we describe a physical system by its position and momentum. Knowing the position and momentum of every billiard ball at a given instant lets us predict the system’s future evolution using Newton’s second law, where F = ma (force equals mass times acceleration).

Quantum mechanics asks the same question but cannot be answered with Newton’s law because position and momentum are no longer the correct variables. Quantum objects do not always behave like tiny billiard balls; sometimes they are better described as waves. Light, for example, was first modeled as particles by Newton, later as waves by Maxwell, and finally as both particles (photons) and waves by Einstein’s photoelectric effect explanation.

Einstein showed that the energy of a photon is proportional to its frequency ( E = hν ), introducing the Planck constant h . This led to the realization that light can exhibit either wave‑like or particle‑like behavior depending on the experiment.

Inspired by this duality, Louis de Broglie hypothesized that matter also has wave‑particle duality: particles such as electrons can behave like waves under certain conditions. Experimental evidence in the 1920s confirmed electron diffraction, supporting de Broglie’s idea.

The most famous demonstration is the double‑slit experiment. When electrons (or photons, neutrons) are fired one at a time through two slits, an interference pattern emerges on a detector screen, even though each particle arrives as a localized impact. This paradoxical result forces us to accept the probabilistic nature of quantum phenomena.

De Broglie’s hypothesis implies that the same wavelength‑momentum relation that applies to photons should apply to any particle. In classical wave theory, the evolution of waves (e.g., sound or water waves) is described by the wave equation, whose solutions give the wave’s shape at any time.

Similarly, a “matter wave” would be described by a wave function that encodes all information about a quantum system. In 1926 Erwin Schrödinger formulated the Schrödinger equation for a single particle in three dimensions:

The equation relates the particle’s kinetic energy, potential energy, and the wave function.

When the potential does not depend on time, the equation separates into a time‑independent part, yielding stationary states with quantized energies. The square of the wave function’s magnitude gives the probability density of finding the particle at a given position (Born’s rule).

The uncertainty principle, discovered by Werner Heisenberg, states that the more precisely we know a particle’s position, the less precisely we can know its momentum, and vice versa. This is not a limitation of measurement devices but a fundamental property of nature, implying that particles do not have definite trajectories.

Schrödinger’s equation successfully explained the discrete energy spectrum of the hydrogen atom, electron behavior in solids (solid‑state physics), and phenomena such as quantum tunneling, where particles can cross energy barriers they classically should not overcome.

These breakthroughs revealed that microscopic particles obey probabilistic wave mechanics rather than deterministic Newtonian dynamics, reshaping our understanding of the physical world.

Reference: Freiberger, M. (2012, August 2). Schrödinger's equation — what is it?