When you throw a ball in the air, the equations of classical physics will tell you exactly what path the ball will take as it falls, and when and where it will land. But if you were to squeeze that same ball down to the size of an atom or smaller, it would behave in ways beyond anything that classical physics can predict.Or so we’ve thought.MIT scientists have now shown that certain mathematical ideas from everyday classical physics can be used to describe the often weird and nonintuitive behavior that occurs at the quantum, subatomic scale.In a paper appearing today in the journal Proceedings of the Royal Society, the team shows that the motion of a quantum object can be calculated by applying an idea from classical physics known as “least action.” With their new formulation, they show they can arrive at exactly the same solution as the Schrödinger equation — the main description of quantum mechanics — for a number of textbook quantum-mechanical scenarios, including the double-slit experiment and quantum tunneling.Such mysterious phenomena, that could only be understood through equations of quantum mechanics, can now also be described using the team’s new classical formulation.
In essence, the researchers have built an exact mathematical bridge between the classical, everyday physical world and the world that happens at dimensions smaller than an atom.“Before, there was a very tenuous bridge that worked only for reasonably large [quantum] 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.”“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.”To infinity and far belowSlotine and Lohmiller derived the quantum bridge while working on solidly classical problems. The researchers are members of the MIT Nonlinear Systems Laboratory, which Slotine directs. He and his colleagues develop models to describe complex behavior in problems of robotic and aircraft control, neuroscience, and machine learning.
To predict the behavior of such systems, engineers often look to the Hamilton-Jacobi equation, which is one of the major formulations of classical mechanics and is related to Newton’s famous laws of motion.The Hamilton-Jacobi equation essentially represents an object’s motion as minimizing a quantity called the action. Take, for instance, a simple scenario in which a ball is thrown from point A to point B. Theoretically, the ball could take any number of zigzagging paths between the two points.
But the equation states that the actual path should be one where the ball’s “action” is minimized at every single point along that path.In this case, the term “action” refers to the sum over time of the difference between an object’s kinetic energy (the energy that is generating the motion) and its potential energy (the object’s stored energy). The actual path that a ball takes between point A and B should then be a sequence of positions where the overall difference between kinetic and potential energy is minimized.Slotine and Lohmiller were applying the Hamilton-Jacobi equation, and the principle of least action, to a number of classical mechanics problems with constraints when they realized that the equation, with some mathematical extensions, could solve a famous problem in quantum mechanics known as the double-slit experiment.The double-slit experiment illustrates one of the weird, nonclassical behaviors that arises at quantum scales.
In the experiment, two slits are cut out of a metal wall. When a single photon — a quantum-scale particle of light — is shot toward the wall, classical physics predicts that you should see a spot of light on the other side of the wall, assuming that the photon flew straight through either one of the holes, following a single path.But experimentalists have instead observed alternating bright and dark stripes. The reality-bending pattern is a result of a quantum mechanical phenomenon by which a photon takes more than one path simultaneously.
In this context, when a single photon is shot toward the wall, it can pass through both holes at the same time, along two paths that end up interfering with each other. The pattern of stripes that results means that the photon’s two interfering paths must be wave-like. The experiment therefore demonstrates how a quantum particle can also behave, however improbably, like a wave.Since the discovery of quantum mechanics, physicists have tried to explain the double-slit experiment using tools from classical, everyday physics. But they’ve only ever been able