Classical shallow water wave systems exhibit quantum behaviors

Quantum mechanics describes the behavior of the universe on tiny scales. However, some research has found that certain quantum behaviors, such as probabilistic outcomes and discrete energy states, can be seen even in classical dynamics.

One example is a wave in shallow water. Though governed by classical hydrodynamics, shallow water waves can have waveforms and dynamics found in quantum realms. Idan Ceausu and Yuval Dagan explored the dynamics of shallow water waves to see if they could be used for modeling quantum-like behavior.

With numerical simulations, Ceausu and Dagan modeled the movement of real-world particles gliding on the slopes of shallow water waves. They computed the particles’ spatial probability distribution and found the particles’ paths could be periodic or chaotic..

“Remarkably, this distribution matched the quantum mechanical predictions for particles in potential wells,” Dagan said. “This demonstrates that quantum-like statistics can emerge from deterministic dynamics, with intermittent chaos and order playing a central role.”

The classical shallow wave system naturally reproduced a quantum principle that states the probability of finding a particle is proportional to the square of the wavefunction. The findings showcase how classical systems can be used to visualize and study foundational quantum mechanics questions, such as wave-particle duality and the origin of probability. The discovery that quantum-like statistics can be encoded in chaotic dynamics is also useful in turbulence modeling, statistical mechanics and quantum-inspired computing.

“It offers a new perspective on quantum probability, suggesting that what appears as intrinsic randomness may instead arise from structured but unpredictable motion within nonlinear systems,” Dagan said.

Source: “Quantum particle statistics in classical shallow water waves,” by Idan Ceausu and Yuval Dagan, Chaos: An Interdisciplinary Journal of Nonlinear Science (2025). The article can be accessed at https://doi.org/10.1063/5.0263305 .