that the phase function for the waves, , in a mechanical system should be identified with the solution to the Hamilton–Jacobi equation, an equation which even William Rowan Hamilton in the 19th century believed to be a short-wavelength limit of a sort of wave mechanics. Schrödinger then found the proper wave equation which matched the Hamilton–Jacobi equation for the phase, this is the famous equation that bears his name. Kramers transition matrix The old quantum theory was formulated only for special mechanical systems which could be separated into action angle variables which were periodic. It did not deal with the emission and absorption of radiation. Nevertheless, Hendrik Kramers was able to find heuristics for describing how emission and absorption should be calculated. Kramers suggested that the orbits of a quantum system should be Fourier analyzed, decomposed into harmonics at multiples of the orbit frequency: The index n describes the quantum numbers of the orbit, it would be n–l–m in the Sommerfeld model. The frequency is the angular frequency of the orbit while k is an index for the Fourier mode. Bohr had suggested that the k-th harmonic of the classical motion correspond to the transition from level n to level n−k. Kramers proposed that the transition between states were analogous to classical emission of radiation, which happens at frequencies at multiples of the orbit frequencies. The rate of emission of radiation is proportional to , as it would be in classical mechanics. The description was approximate, since the Fourier components did not have frequencies that exactly match the energy spacings between levels. This idea led to the development of matrix mechanics. Limitations The old quantum theory had some limitations: The old quantum theory provides no means to calculate the intensities of the spectral lines. It fails to explain the anomalous Zeeman effect (that is, where the spin of the electron cannot be neglected). It cannot quantize "chaotic" systems, i.e. dynamical systems in which trajectories are neither closed nor periodic and whose analytical form does not exist. This presents a problem for systems as simple as a 2-electron atom which is classically chaotic analogously to the famous gravitational three-body problem. However it can be used to describe atoms with more than one electron (e.g. Helium) and the Zeeman effect. It was later proposed that the old quantum theory is in fact the semi-classical approximation to the canonical quantum mechanics but its