When you throw a ball into the air, the equations of classical physics tell you exactly what path it will take as it falls, and exactly when and where it will land. However, if you squeeze the same ball down to a subatomic size, it will behave in ways that cannot be predicted by classical physics.
Or so we thought.
Scientists at MIT have shown that certain mathematical ideas from everyday classical physics can be used to explain often strange and unintuitive behavior that occurs at the quantum, subatomic scale.
In a paper published in today’s journal Proceedings of the Royal Societythe research team showed that the motion of quantum objects can be calculated by applying an idea from classical physics known as “least action.” They showed that their new formulation allows them to reach exactly the same solutions as the Schrödinger equation, the main description of quantum mechanics, for many textbook quantum mechanics scenarios, such as double-slit experiments and quantum tunneling.
Such mysterious phenomena could only be understood through the equations of quantum mechanics, but now they can also be explained using the team’s new classical formulation. In essence, the researchers have built a precise mathematical bridge between the classical everyday physical world and the world that takes place in subatomic dimensions.
“Before, we had very weak bridges that only worked for fairly large bridges. [quantum] “Now we have a powerful bridge: a common way to describe quantum mechanics, classical mechanics, and relativity that works at all scales,” said study co-author Winfried Lohmiller, a researcher in the Massachusetts Institute of Technology’s Nonlinear Systems Laboratory.
“I’m not saying there’s anything wrong with quantum mechanics,” emphasizes co-author Jean-Jacques Slotine, a professor of mechanical engineering and information science, and of brain and cognitive sciences at the Massachusetts Institute of Technology. “We’re just showing a different way of calculating quantum mechanics, which is based on well-known classical ideas and summarizes them in a simple way.”
To infinity and far below
Slotine and Lohmiller arrived at the quantum bridge by working on a solid classical problem. The researchers are members of the MIT Nonlinear Systems Laboratory, of which Slotine is director. He and his colleagues develop models to describe complex behavior in robot and aircraft control, neuroscience, and machine learning problems. To predict the behavior of such systems, engineers often turn to the Hamilton-Jacobi equation. This is one of the main formulations of classical mechanics and is related to Newton’s famous laws of motion.
The Hamilton-Jacobi equation essentially describes the motion of an object as minimizing a quantity called an action. For example, consider a simple scenario where a ball is thrown from point A to point B. In theory, a ball can take any number of zigzag paths between two points. However, the equations show that the actual trajectory must be such that the ball’s “motion” is minimized at every point along its trajectory.
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 producing its motion) and its potential energy (the object’s stored energy). The actual path that the ball follows between points A and B must be a series of positions where the overall difference between kinetic and potential energy is minimized.
As Slottin and Lohmiller applied the Hamilton-Jacobi equation and the principle of least action to a number of constrained classical mechanics problems, they realized that a mathematical extension of the equation could solve a famous problem in quantum mechanics known as the double-slit experiment.
The double-slit experiment illustrates one of the strange non-classical behaviors that occur at the quantum scale. In the experiment, two slits are cut out in a metal wall. When a single photon, a quantum-scale particle of light, is fired toward a wall, classical physics predicts that, assuming the photon follows a single path straight through either hole, we should see a point of light on the other side of the wall.
However, the experimenters instead observed alternating light and dark stripes. Reality-altering patterns are the result of a quantum mechanical phenomenon in which photons take multiple paths at the same time. In this context, when a single photon is fired toward a wall, it can pass through both holes simultaneously and along two paths that eventually interfere with each other. The resulting fringe pattern means that the photon’s two interference paths must be wavy. This experiment thus demonstrates that quantum particles can also behave like waves, although this is unlikely.
Since the discovery of quantum mechanics, physicists have attempted to explain double-slit experiments using the tools of classical everyday physics. However, they were only able to approximate the experimental results.
Even eminent physicist Richard Feynman ’39 thought this task was impossible. He assumed that all the theoretical paths that a photon could take, whether it be a straight line or a variation of a zigzag path through either of the two holes, would have to be considered and averaged. Such an exercise requires computing an infinite number of possible zigzag paths, all of which contradict the expected classical smooth path.
This last point is something Slotine and Lohmiller realized they could adjust to. While classical physics assumes that objects must take only one path from point A to B, quantum mechanics allows objects to take multiple paths and multiple states at the same time. This is a fundamental quantum property known as superposition.
The research team thought as follows. What if classical physics could also handle this concept of multiple paths, at least mathematically? And they reasoned that there was no need to calculate an infinite number of paths. Instead, a much smaller number of “minimal action” classical paths could produce exactly the same quantum result.
With this idea in mind, they looked back at the Hamilton-Jacobi equation and considered how its least-action principle could be applied to predict double-slit experiments and other quantum phenomena.
“For a while, we thought it was too good to be true,” Sloteen says.
The fate of a particle lies in its density
In their new study, the research team added another element of classical physics: density. This is essentially the probability that a given path will be followed.
“We think of density in terms of fluid mechanics,” Lohmiller explains. “In the double-slit experiment, imagine pumping a hose against a wall. What happens is that most of the water hits the center, but some droplets also go to the sides. The higher density of water in the center means that there’s a higher chance of finding a droplet along that path. And since there is a distribution, we can calculate it.”
He and Slotine tweaked the Hamilton-Jacobi equation to include a density term and multiple paths of minimum action and applied it to a double-slit experiment. They found that compared to Feynman’s infinite zigzag path, this formulation only requires considering two classical paths through two slits. Ultimately, their classical density and action calculations produced a wave function, or distribution of the paths a photon is most likely to take. This was exactly what was predicted by the Schrödinger equation, the central equation used to describe quantum mechanical behavior.
“We showed that if you calculate the density properly, the Schrödinger equation of quantum mechanics and the Hamilton-Jacobi equation of classical physics are actually identical,” Slottin says. “This is a purely mathematical result. We are not saying that quantum phenomena occur on classical scales. We are saying that this quantum behavior can be calculated with very simple classical tools.”
In addition to double-slit experiments, the researchers showed that the reworked equations can also predict other quantum mechanical behaviors, such as quantum tunneling, which allows particles such as electrons to pass through energy barriers not possible in classical physics. They were also able to derive the precise quantum waves of electrons in hydrogen atoms from the planet’s classical orbit. Finally, they reconsidered the famous Einstein-Podolsky-Rosen experiment that started the modern study of quantum entanglement in this light.
The researchers envision that scientists can use the new formula as a simple way to predict how a particular quantum system or device will behave.
“This could have important implications for quantum computing, where qubits have nonlinear energies that physicists have to approximate, and for a deeper understanding of issues related to both quantum physics and general relativity,” Slotin suggests. “At least in principle, we should be able to precisely characterize this quantum behavior using simple classical tools and show that it’s not so mysterious after all.”
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