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Quantum mechanics and classical physics have been at war for a long time. It turns out they've been allies all along.
MIT physicists have proposed a new way to look at one of the oldest boundaries in science—the boundary between classical and quantum physics. These two fields typically describe the world in very different languages. Classical physics works well when objects move along clear trajectories and obey familiar intuitions. Quantum mechanics begins when particles behave strangely: they can take multiple paths at once, interfere with themselves, and obey rules that don't fit well with conventional notions of motion. The authors of the new paper don't claim that this boundary has disappeared. They demonstrate something else: some quantum effects can be achieved using the tools of classical physics, given the right mathematical framework.
This is precisely what makes this work so interesting. For a long time, classical and quantum physics were perceived as two different ways of describing reality, whose results converge only in a few special cases but do not directly transition from one to the other. In the classical picture, a particle typically has a single trajectory, while in quantum physics, one must work with probabilities, wave functions, and superpositions of states. Because of this difference, it seemed that between the two theories there lay not just a technical gap, but a deeper mathematical chasm. MIT physicists have attempted to show that, at least for some problems, this gap is not as great as it seemed.
The work was carried out by Winfried Lohmiller and Jean-Jacques Slotin of the MIT Nonlinear Systems Laboratory. They demonstrated that, when the problem is formulated in a certain way, the classical Hamilton-Jacobi equation leads to the same results as the Schrödinger equation , which underlies quantum mechanics. The authors tested this approach on several well-known examples, including the double-slit experiment and quantum tunneling.
The simplest way to understand the concept is with a double-slit experiment. Two narrow openings are made in an opaque partition and, for example, single photons are directed toward them. If we look at the problem from the perspective of classical physics, the particle should pass through either the left slit or the right. Then, on the screen behind the partition, a pattern resembling the sum of two independent beams would appear. But in a real experiment, alternating light and dark bands appear. This interference pattern demonstrates that a simple explanation in terms of a single chosen path is insufficient.
This result is usually explained through quantum superposition . The photon behaves as if it were traveling along several paths at once, and these paths then influence each other. This is why a wave pattern arises, even though we are talking about a particle. Richard Feynman described this effect by summing over all possible trajectories. In such a picture, one must consider not only direct paths, but all possible variations, including the most complex and complex ones. This is where quantum mechanics begins to sharply diverge from familiar classical intuition.
The authors of the new paper took a different approach. They asked whether the same result could be achieved without trying an infinite number of trajectories. They based their work on the Hamilton-Jacobi equation. In classical mechanics, it is related to the principle of least action. This principle states that if a body moves from point A to point B, its motion can be described by a quantity that depends on the energy of the system, and among the possible paths, the one for which this quantity is minimal is selected. In the case of a thrown ball, this refers to the relationship between kinetic and potential energy throughout the entire motion.
The researchers then added a density function to this scheme and rewrote the problem for the double-slit experiment. This step proved crucial. Instead of an infinite set of possible trajectories, it was sufficient to consider just two classical paths passing through both slits. This calculation yielded a wave function that describes the distribution of possible photon paths and matches the prediction of the Schrödinger equation.
This is the main result of the work. According to the authors, the Schrödinger equation and the Hamilton-Jacobi equation are mathematically identical if the density is calculated correctly. It is important not to oversimplify the meaning to the point of misleading conclusions. The researchers do not claim that quantum effects in the ordinary world are directly governed by classical physics. They demonstrate that in some cases, quantum behavior can be calculated using simpler classical methods, provided the appropriate mathematical framework is used.
If this approach proves useful in more complex problems, physicists will have a new tool for working with quantum systems. For now, the authors have demonstrated it using well-known educational and theoretical examples, but this alone was enough to attract attention. Quantum mechanics remains a distinct field with its own laws, but the mathematical connection between it and classical physics is now noticeably clearer.
MIT physicists have proposed a new way to look at one of the oldest boundaries in science—the boundary between classical and quantum physics. These two fields typically describe the world in very different languages. Classical physics works well when objects move along clear trajectories and obey familiar intuitions. Quantum mechanics begins when particles behave strangely: they can take multiple paths at once, interfere with themselves, and obey rules that don't fit well with conventional notions of motion. The authors of the new paper don't claim that this boundary has disappeared. They demonstrate something else: some quantum effects can be achieved using the tools of classical physics, given the right mathematical framework.
This is precisely what makes this work so interesting. For a long time, classical and quantum physics were perceived as two different ways of describing reality, whose results converge only in a few special cases but do not directly transition from one to the other. In the classical picture, a particle typically has a single trajectory, while in quantum physics, one must work with probabilities, wave functions, and superpositions of states. Because of this difference, it seemed that between the two theories there lay not just a technical gap, but a deeper mathematical chasm. MIT physicists have attempted to show that, at least for some problems, this gap is not as great as it seemed.
The work was carried out by Winfried Lohmiller and Jean-Jacques Slotin of the MIT Nonlinear Systems Laboratory. They demonstrated that, when the problem is formulated in a certain way, the classical Hamilton-Jacobi equation leads to the same results as the Schrödinger equation , which underlies quantum mechanics. The authors tested this approach on several well-known examples, including the double-slit experiment and quantum tunneling.
The simplest way to understand the concept is with a double-slit experiment. Two narrow openings are made in an opaque partition and, for example, single photons are directed toward them. If we look at the problem from the perspective of classical physics, the particle should pass through either the left slit or the right. Then, on the screen behind the partition, a pattern resembling the sum of two independent beams would appear. But in a real experiment, alternating light and dark bands appear. This interference pattern demonstrates that a simple explanation in terms of a single chosen path is insufficient.
This result is usually explained through quantum superposition . The photon behaves as if it were traveling along several paths at once, and these paths then influence each other. This is why a wave pattern arises, even though we are talking about a particle. Richard Feynman described this effect by summing over all possible trajectories. In such a picture, one must consider not only direct paths, but all possible variations, including the most complex and complex ones. This is where quantum mechanics begins to sharply diverge from familiar classical intuition.
The authors of the new paper took a different approach. They asked whether the same result could be achieved without trying an infinite number of trajectories. They based their work on the Hamilton-Jacobi equation. In classical mechanics, it is related to the principle of least action. This principle states that if a body moves from point A to point B, its motion can be described by a quantity that depends on the energy of the system, and among the possible paths, the one for which this quantity is minimal is selected. In the case of a thrown ball, this refers to the relationship between kinetic and potential energy throughout the entire motion.
The researchers then added a density function to this scheme and rewrote the problem for the double-slit experiment. This step proved crucial. Instead of an infinite set of possible trajectories, it was sufficient to consider just two classical paths passing through both slits. This calculation yielded a wave function that describes the distribution of possible photon paths and matches the prediction of the Schrödinger equation.
This is the main result of the work. According to the authors, the Schrödinger equation and the Hamilton-Jacobi equation are mathematically identical if the density is calculated correctly. It is important not to oversimplify the meaning to the point of misleading conclusions. The researchers do not claim that quantum effects in the ordinary world are directly governed by classical physics. They demonstrate that in some cases, quantum behavior can be calculated using simpler classical methods, provided the appropriate mathematical framework is used.
If this approach proves useful in more complex problems, physicists will have a new tool for working with quantum systems. For now, the authors have demonstrated it using well-known educational and theoretical examples, but this alone was enough to attract attention. Quantum mechanics remains a distinct field with its own laws, but the mathematical connection between it and classical physics is now noticeably clearer.
