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 below
Slotine 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 to approximate the experiment's results.
Even the noted physicist Richard Feynman '39 found the task impossible. He assumed that one would have to consider and average over every single theoretical path that a photon could take, whether it be a straight line or any variation of a zigzagging path through either of the two holes. Such an exercise would require calculating an infinite number of possible zigzag paths, which all contradict the classical smooth paths one would expect.
This last point is what Slotine and Lohmiller realized could be tweaked. Where classical physics assumes that an object must only take a single path from point A to B, quantum mechanics allows for an object to take multiple paths and multiple states simultaneously - a fundamental quantum property known as superposition.
The team wondered: What if classical physics could also entertain, at least mathematically, this notion of multiple paths? Then, they reasoned that an infinite number of paths wouldn't have to be calculated. Instead, a much smaller number of "least action" classical paths might produce the exact same quantum result.
With this idea in mind, they looked back to the Hamilton-Jacobi equation to see how they might adapt its principles of least action to predict the double-slit experiment and other quantum phenomena.
"For a while we thought it was a little too good to be true," Slotine says.
A particle's destiny is in its density
In their new study, the team adds another ingredient of classical physics: "density," which is, essentially, a probability that a given path is taken.
"We think of density in terms of fluid dynamics," Lohmiller explains. "For the double-slit experiment, imagine pumping a hose toward the wall. What will happen is, most of the water will hit the center, but some droplets will also go toward the sides. 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."
He and Slotine tweaked the Hamilton-Jacobi equation to include terms of density and multiple least action paths, and applied it to the double-slit experiment. They found that with this formulation, they only had to consider two classical paths through the two slits, as compared to Feynman's infinity of zigzag paths. Ultimately, their calculations of classical density and action produced a wave function, or distribution of most probable paths that a photon could take, that was exactly the same as what was predicted by the Schrödinger equation, which is the central equation used to describe quantum-mechanical behavior.
"We show that the Schrödinger's equation of quantum mechanics and the Hamilton-Jacobi equation of classical physics are actually identical given a suitable computation of density," Slotine says. "That's a purely mathematical result. We're not saying that quantum phenomena happens at classical scales. We're saying you can compute this quantum behavior with very simple classical tools."
In addition to the double-slit experiment, the researchers showed the reworked equation can also predict other quantum mechanical behavior, such as quantum tunneling, in which particles such as electrons can pass through energy barriers that would not be possible according to classical physics. They could also derive the exact quantum wave of the electron in a hydrogen atom from the classical orbit of a planet. Finally, they revisited from this perspective the famous Einstein-Podolski-Rosen experiment, which started the modern study of quantum entanglement.
The researchers envision that scientists could use the new formula as a simple method to predict how certain quantum systems and devices will perform.
"There could be important implications for quantum computing, where quantum bits have these nonlinear energies that physicists must approximate, or for better understanding problems involving both quantum physics and general relativity," Slotine offers. "In principle at least, we should now be able to characterize this quantum behavior exactly, with simple classical tools, and show that it's not so mysterious after all."
