Modern quantum mechanics began with a simple, yet deeply mysterious, assumption: particles can behave like waves, and waves can behave like particles. In 1923, Louis de Broglie proposed an elegant mathematical formulation of this idea: every moving particle carries with it a wave-like nature, characterised by a wavelength given by the now-famous relation λ = h/mv, where λ is the wavelength, m the particle’s mass, v its velocity, and h the Planck constant. This was a revolutionary idea—one that linked the seemingly disjoint worlds of particles and waves through a single mathematical expression. It didn’t take long for de Broglie’s proposal to find experimental support. In the 1920s and 1930s, electron diffraction experiments showed that beams of electrons passing through crystals formed interference patterns—just like light waves. These results left no doubt: matter truly exhibits wave-like properties.
De Broglie’s relation not only changed how we think about particles; it also inspired the next major step in quantum theory. Erwin Schrödinger explicitly credited de Broglie’s work as the inspiration for what became known as the Schrödinger equation, which represents the cornerstone of modern quantum physics. Interestingly, the statistical interpretation of the Schrödinger equation—the link between its wave function and measurement outcomes—was not developed by Schrödinger himself, but by Max Born. The Born rule states that the square of the wave function’s amplitude gives the probability of finding a particle at a particular location. Soon, most physicists came to believe that quantum theory was fundamentally different from classical statistical mechanics: that it was inherently random, and that no underlying deterministic theory based on well-defined particle trajectories could reproduce its predictions.
In the 1950s, David Bohm challenged this assumption by developing an alternative theory of quantum phenomena, demonstrating that what was widely thought to be impossible was, in fact, possible. In what is now known as Bohmian mechanics—also referred to as the de Broglie–Bohm theory or pilot-wave theory—particles follow definite paths, guided by the quantum wave function. Bohmian mechanics offers a clear picture of where a particle is at any given time and how it moves from one point to another.
The mathematical instrument in Bohmian mechanics that leads to well-defined particle trajectories is the guiding equation, which is postulated in addition to the Schrödinger equation. Using the wave function provided by the Schrödinger equation, the guiding equation defines a velocity field for the particles. This velocity is proportional to the gradient of the complex phase of the wave function. For a stream of moving particles described by a traveling wave of the form exp(ikx), for example—where k is the wavenumber and i is the imaginary unit—the velocity is ℏk/m. In contrast, for a standing wave pattern described by cos(kx), the velocity is expected to be zero, since there is no complex phase associated with this wave function. The same holds for an evanescent wave function of the form exp(–x/λ), as it occurs in tunnelling or reflection processes.
This is precisely where our work comes in. We investigate the particle motion associated with evanescent wave functions, which appear when particles are reflected at a potential step. To do this, we confine the particles to one-dimensional waveguides—similar to how optical fibers guide light—and couple two such waveguides in a controlled manner by placing them parallel to each other at a close distance. The coupling between the waveguides provides a temporal reference for the particles’ motion. By comparing the motion of particles within a waveguide to the tunneling-induced hopping between the waveguides, we can draw conclusions about how fast the particles move. This experimental approach is somewhat analogous to the physics of a horizontal throw. If a stone is thrown from a cliff, for example, it follows a parabolic trajectory (neglecting air resistance), as its motion combines a uniform horizontal component with accelerated vertical motion due to gravity. The curvature of the parabola can be interpreted as a measure of the horizontal speed, assuming gravity is known. In a similar way, we interpret the population transfer between the coupled waveguides as a measure of the speed along the waveguide axis in our experiments.
The outcome of our experiments is that particles move where they are not expected to move. Contrary to the predictions of the Bohmian guiding equation, our experimental signatures indicate that particles in evanescent quantum states are not at rest—they move with a well-defined speed. The measured speed supports the core idea behind de Broglie’s relation: that motion and wavelength are fundamentally linked—albeit with a twist. For evanescent wave functions, we find that a de Broglie relation of the form λ = ℏ/mv holds, where λ is the decay length and v represents a non-directional particle speed. Thus, our results suggest that phase gradients and amplitude gradients play complementary roles in indicating motion within a quantum mechanical wave function. This stands in contrast to the guiding equation in Bohmian mechanics, which attributes motion solely to phase gradients. In other words, our findings call into question whether the ontology implied by Bohmian trajectories is realised in nature.
The intellectual lineage from de Broglie to Bohm reflects a deeper human aspiration to understand quantum mechanics—not just use it. But such aspirations have not always been welcomed in the scientific mainstream. David Bohm himself faced harsh criticism for his theory of quantum phenomena. Opposition to his idea was often based on conceptual discomfort rather than rigorous counterarguments. In the decades that followed, the dominant trend in physics moved away from foundational questions, with a growing emphasis on practical applications and technical advances—an attitude often captured by the phrase “shut up and calculate.”
Although the situation has improved in recent years, key decision-makers in universities and the scientific community still too often fail to recognise the importance of addressing foundational questions. This stands in stark contrast to the widespread interest these questions generate among the general public. Many of the deepest mysteries of quantum mechanics—such as what exactly constitutes a measurement—remain unresolved. Yet these questions lie at the very heart of how we understand nature, and how we design and interpret the outcomes of our most advanced experiments. Progress in this area can lead to major breakthroughs in knowledge, potentially redefining what is technically achievable and which kinds of applications are even conceivable.