The mathematical approximation method which, since the breakthrough of quantum mechanics, has usually been called the WKB method, has in reality been known for a very long time. The method describes various kinds of wave motion in an inhomogeneous medium, where the properties change only slightly over one wavelength, and it also provides the connection between classical mechanics and quantum mechanics. To a surprisingly large extent it can already be found in an investigation by Carlini (1817) on the motion of a planet in an unperturbed elliptic orbit. After that the method was independently developed and used by many people. The important connection formulas were, however, missing, until Rayleigh (1912) very implicitly and Gans (1915) somewhat more explicitly derived one of them, which was later rediscovered independently by Jeffreys (1925), who also derived another connection formula (although not in quite the correct form), and by Kramers (1926).

Development from 1817 to 1926

Carlini's pioneering work

At the beginning of the nineteenth century Carlini (1817) (Fig. 1.1.1) treated an important problem in celestial mechanics. He considered the motion of a planet in an elliptic orbit around the sun, with the perturbations from all other gravitating bodies neglected. Using a polar coordinate system in the plane of the planetary motion, with the origin at the sun, one can express the polar angle as 2πt/T plus an infinite series containing sines of integer multiples of 2πt/T, where t is the time counted from a perihelion passage, i.e., from a moment when the planet is closest to the sun, and T is the time for one revolution of the planet in its orbit.