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How does the correspondence principle connect quantum mechanics with classical mechanics?

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A diagram of a cylindrical vessel containing gas, with a movable piston held down by a weight. Countless small dots beneath the piston represent gas particles in rapid, random motion, illustrating Bernoulli's model of gas pressure. — physics As conceived by Daniel Bernoulli in *Hydrodynamica* (1738), gases consist of numerous particles in rapid random motion. He assumed that the pressure of a gas is produced by the direct impact of the particles on the walls of the container.

The correspondence principle, formulated by Niels Bohr in 1923, serves as a vital bridge between quantum mechanics and classical physics. It posits that any new physical theory must account for the phenomena explained by its predecessor, ensuring that the new framework reproduces the results of classical physics when applied to conventional, macroscopic scales.1

In the context of atomic systems, the principle demonstrates that quantum descriptions of electron transitions align with classical electromagnetic theory as quantum numbers become large. While classical physics fails to explain the stability of atoms—predicting that orbiting electrons should lose energy and collapse—the correspondence principle ensures that quantum theory remains consistent with the laws of daily experience in the limit where quantum effects become negligible. This requirement acts as a philosophical and mathematical guideline, ensuring continuity in our understanding of the physical world across different scales.21