Researchers Calculate Quantum Motion Using Classical “Least Action”

Researchers at MIT have demonstrated that the motions of quantum particles can be mathematically replicated using the classical principle of “least action”, the same concept used to predict the trajectory of a thrown ball. The team successfully used this classical approach to calculate the motion of quantum objects, achieving the same solutions as the complex Schrödinger equation for several textbook quantum-mechanical scenarios, including the double-slit experiment and quantum tunneling. This work establishes a mathematical connection between classical and quantum mechanics, creating what researchers call a “strong bridge” between the two realms. “Before, there was a tenuous bridge that worked only for reasonably large particles,” says study co-author Winfried Lohmiller, a research associate in the Nonlinear Systems Laboratory at MIT. “Now we have a strong bridge—a common way to describe quantum mechanics, classical mechanics, and relativity, that holds at all scales.”

Classical Least Action Mirrors Quantum Motion

A principle governing the motion of everyday objects can now accurately model the behavior of particles at the quantum level, challenging long-held assumptions about the divide between classical and quantum physics. This mathematical connection suggests a new approach to understanding quantum behavior. The team, working within the MIT Nonlinear Systems Laboratory, arrived at this connection while investigating classical mechanics problems involving constraints. They focused on the Hamilton-Jacobi equation, a formulation of classical mechanics related to Newton’s laws, which states that an object’s motion minimizes a quantity called “action.” Considering a ball thrown from point A to point B, the equation dictates the actual path minimizes the sum of kinetic and potential energy at every point. The researchers realized that extending this principle, with careful mathematical adjustments, could accurately model the double-slit experiment, a hallmark of quantum mechanics where particles exhibit wave-like behavior.

Traditionally, explaining the double-slit experiment required considering many least action paths a particle could take, a computationally intensive and conceptually challenging task. Physicist Richard Feynman ’39 famously grappled with this issue, finding it impossible to approximate the results without averaging over all theoretical paths. Slotine and Lohmiller’s breakthrough lies in recognizing that quantum mechanics allows for multiple paths and states simultaneously, a property known as superposition, and adapting the classical least action principle to accommodate this. “We’re not saying there’s anything wrong with quantum mechanics,” emphasizes co-author Jean-Jacques Slotine, an MIT professor of mechanical engineering and information sciences, and of brain and cognitive sciences. “We’re just showing a different way to compute quantum mechanics, which is based on well-known classical ideas that we put together in a simple way.”

Hamilton-Jacobi Equation Extended to Quantum Scales

The longstanding challenge of reconciling classical and quantum physics has yielded a surprising development; researchers at MIT have demonstrated a mathematical bridge between the classical Hamilton-Jacobi equation and the Schrödinger equation, the cornerstone of quantum mechanics, for several textbook quantum-mechanical scenarios. While quantum mechanics accurately describes the subatomic world, its foundations remain conceptually distinct from the deterministic framework of classical physics, leading to decades of effort to bridge the gap between the two. Previous attempts to connect these realms often relied on approximations or limited applicability, but this new formulation offers a precise mathematical correspondence. The key innovation lies in adapting the classical equation to accommodate the quantum principle of superposition, where a particle can exist in multiple states simultaneously.

Rather than calculating an infinite number of possible paths, as previously required to model quantum behavior, the team showed that a smaller number of “least action” classical paths might produce the same result. “We think of density in terms of fluid dynamics,” Lohmiller explains. “For the double-slit experiment, imagine pumping a hose toward the wall. Most of the water will hit the center, but some droplets will also go toward the sides.” This approach allowed them to derive the same wave function predicted by the Schrödinger equation, even for scenarios like quantum tunneling.

Double-Slit Experiment Solved with Classical Paths

Researchers at MIT are challenging conventional wisdom regarding the foundations of quantum mechanics, demonstrating a mathematical bridge between classical and quantum descriptions of particle behavior. The team, operating within the Nonlinear Systems Laboratory, has successfully applied the classical “least action” principle, a cornerstone of predicting trajectories for macroscopic objects, to accurately model quantum phenomena, including the famed double-slit experiment and quantum tunneling. This achievement bypasses the need for complex quantum calculations in several textbook quantum-mechanical scenarios, offering a new pathway for understanding the subatomic world. The team discovered that by considering a smaller number of “least action” paths, weighted by their probability density, they could replicate the results traditionally obtained using the Schrödinger equation, the fundamental equation governing quantum mechanics. This wasn’t merely an approximation; the classical formulation yielded identical solutions for the tested quantum scenarios.

This approach directly addresses a long-standing challenge in physics: explaining the double-slit experiment without invoking inherently quantum concepts. Historically, physicists have struggled to reconcile the observed wave-like behavior of particles in this experiment with classical notions of particle trajectories. Even Richard Feynman ’39 famously found it impossible to explain without considering an infinite number of paths.

Density Integration Bridges Classical & Quantum Probability

The ability to accurately model quantum phenomena using classical physics principles promises advancements in fields ranging from quantum computing to our understanding of gravity, according to new research from MIT. While quantum mechanics routinely predicts outcomes at the subatomic level, interpreting how these outcomes arise has long presented a challenge; now, scientists are demonstrating a surprising mathematical bridge between the behavior described by the Schrödinger equation and the classical principle of least action. This isn’t merely a conceptual link, but a demonstrable ability to compute quantum mechanics using established classical methods, potentially simplifying complex calculations and offering new avenues for technological development. Researchers successfully leveraged the “least action” principle, the idea that physical systems evolve to minimize a quantity called action, to replicate solutions previously obtained through the Schrödinger equation for several textbook quantum-mechanical scenarios, including the double-slit experiment and quantum tunneling.

This innovative approach circumvents the need to calculate an infinite number of potential paths, a major hurdle in previous attempts to bridge classical and quantum descriptions, as famously noted by physicist Richard Feynman. A high density of water at the center means there is a high probability of finding a droplet along that path, and there will be a distribution which we can compute. The researchers demonstrated this approach for several textbook quantum-mechanical scenarios, including the double-slit experiment and quantum tunneling.