When a ball is tossed into the air, classical physics equations can predict its trajectory and landing spot. However, if the ball were reduced to the size of an atom, its behavior would diverge from classical predictions. MIT researchers have demonstrated that certain classical physics concepts can describe the strange behaviors at the quantum level.<\/p>\n
A newly published paper in the Proceedings of the Royal Society reveals that the motion of quantum objects can be calculated using the classical physics notion of “least action.” This new method produces the same results as the Schr\u00f6dinger equation, the core of quantum mechanics, for scenarios like the double-slit experiment and quantum tunneling. The researchers have effectively linked classical and quantum physics with this mathematical approach.<\/p>\n
“Previously, the connection between classical and quantum physics was weak and only applicable to larger quantum particles,” stated Winfried Lohmiller, a co-author and research associate at MIT’s Nonlinear Systems Laboratory. “Now, we’ve established a robust bridge applicable across all scales.” Jean-Jacques Slotine, another co-author and MIT professor, clarified that they are not disputing quantum mechanics but offering an alternative computation method based on classical ideas.<\/p>\n
Slotine and Lohmiller discovered this quantum bridge while working on classical problems. As part of the MIT Nonlinear Systems Laboratory, they develop models for complex systems in fields like robotics and neuroscience. The Hamilton-Jacobi equation, a classical mechanics principle related to Newton’s laws, is central to their work. It describes an object’s motion by minimizing “action,” the difference between kinetic and potential energy over time.<\/p>\n
While applying the Hamilton-Jacobi equation to classical problems, the researchers realized it could solve the quantum double-slit experiment. This experiment demonstrates nonclassical behaviors at the quantum scale. Instead of following a single path, a photon can take multiple paths simultaneously, creating an interference pattern of light and dark stripes.<\/p>\n
Physicists have long struggled to use classical physics to explain the double-slit experiment, only approximating the results. Even Richard Feynman noted the difficulty, as classical assumptions require averaging over infinite possible paths. Slotine and Lohmiller adjusted the classical approach to entertain multiple paths, reducing the needed calculations.<\/p>\n
Their adaptation of the Hamilton-Jacobi equation incorporated “density,” a classical physics concept, to calculate the probability of paths. In the double-slit experiment, this approach required only two classical paths through the slits, aligning with the Schr\u00f6dinger equation’s predictions. “We show that the Schr\u00f6dinger equation of quantum mechanics and the Hamilton-Jacobi equation of classical physics are actually identical given a suitable adaptation,” Slotine and Lohmiller concluded.<\/p>\n