# Jordi, Marta; Perez, Enric: *The Ehrenfest Adiabatic Hypothesis and the Old Quantum Theory, Before Bohr*

In 1918, Bohr published the first two parts of On the quantum theory of line spectra. In this extensive contribution, he offered a quantum theory much more elaborated than the one contained in his seminal works of 1913. Besides innovations of his own, as for example perturbation theory, Bohr incorporated many of the crucial discoveries that had taken place in the years in between: among others, the Einstein’s transition probabilities, the quantum rules of Sommerfeld-Epstein-Schwarzschild and the Ehrenfest adiabatic hypothesis.

The latter one was, no doubt, the one that had had fewer success among Bohr’s colleagues, and, in 1918, it was hardly mentioned. Therefore, until then, its role in the development of the quantum theory had been rather scarce.

In previous works, one of us has studied both the precedents and the genesis of this Ehrenfest’s contribution. Now, in a new paper, we will discuss in detail the content of this hypothesis, focusing our attention especially on the paper that Ehrenfest published in 1916 and 1917 in three different magazines. We will also briefly revise the precedents, which must be located in an exhaustive analysis of Planck’s radiation theory, and the surprising results Ehrenfest achieved in the period 1911-1914, mainly ―but not only― inspired in his first utilization of adiabatic invariance. Some of these results are: the proof of the necessity to introduce discontinuities in the statistical weight function of the black-body system, the strict separation between the statistical characteristics of Einstein’s and Planck’s quanta, and the curve for the specific heat of hydrogen, obtained from the quantization of the rotation energy of diatomic molecules, which was supplying acceptable results at low temperatures. Ehrenfest also managed to relate his hypothesis to the generalization of Boltzmann’s principle, that is, to the statistical interpretation of the second principle of thermodynamics that his teacher Boltzmann had proposed in the previous century.

When Bohr incorporated the adiabatic hypothesis into his theory under the name of ‘principle of mechanical transformability’, he removed its statistical reminiscences, and stressed that the purpose of this principle in his theory was not to establish links to classical mechanics, but simply to guarantee the stability of the quantum stationary states. Besides, the Danish physicist not only overcame some of the obstacles encountered by Ehrenfest, but he even extracted profit of them. Good examples are the ‘singular movements’, which Bohr related to the change of the degree of degeneracy of a system.

Our work will deal with the state of the adiabatic hypothesis of Ehrenfest (its formulation, principal faults, applications, etc.) before Bohr’s intervention, and will be mainly based on a careful analysis of Ehrenfest’s notebooks, correspondence and publications.