Editing Old quantum theory (section)

== 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:
: <math>
X_n(t) = \sum_{k=-\infty}^{\infty} e^{ik\omega t} X_{n;k}
</math>

The index ''n'' describes the quantum numbers of the orbit, it would be ''n''–''l''–''m'' in the Sommerfeld model. The frequency <math>\omega</math> is the angular frequency of the orbit <math>2\pi/T_n</math> 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  <math>|X_k|^2</math>, 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.