Can we still use classical mechanics in the age of quantum mechanics?

The "old quantum theory" of Niels Bohr featured its triumph in 1913 with the theoretical derivation of the energy spectrum of the hydrogen atom, yielding the formula that Johannes Balmer had found empirically in 1885 for a series of absorption lines in the sun's light, and that had been generalized by Rydberg. The theory of Bohr (and Sommerfeld) failed, however, to explain the spectrum of the helium atom. As it was later realized, this was due to the fact that the helium atom classically represents a non-integrable three-body system. After the advent and the triumph of the new "quantum mechanics" of Heisenberg, Schrödinger, and others, in the later 1920ies, the helium problem could eventually be solved, and the old quantum theory was practically forgotten.

Non-integrable classical systems can exhibit chaotic dynamics. Henri Poincaré was the first to show that the solutions of the world's oldest three-body system, Sun-Earth-Moon, could not be obtained in a mathematically convergent form, and that no safe predictions could be made for its stability. With his work published around 1890, Poincaré laid the theoretical foundations of modern chaos research.

Since about 30 years, the original ideas of Bohr and Sommerfeld have received a renaissance in the form of the "periodic orbit theory" developed by Martin Gutzwiller. This theory allows us today, at least partially, to formulate a "semiclassical" quantization of non-integrable and, notably, of chaotic systems. It provides an important instrument for investigating the quantum-mechanical implications of classical chaos, in the quest of the so-called "quantum chaos".