Adiabatic invariants in quantum mechanics

Wolfgang R. Wasow

L. M. Garrido, F.J. Sancho

F. Leyvraz, T. H. Seligman

E. Gozzi

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

In old quantum mechanics the adiabatic theorem established by Ehrenfest had the meaning that the quantized action variables J = n are invariant relative to an infinitely slow (adiabatic) change of a mechanical system.1 From this one can guess that if the system is in a state with definite quantum numbers before the adiabatic change started, then after the changes it will be characterized by the same quantum numbers.

Mário J. de Oliveira

We examine the development of the concept of parametric invariance in classical mechanics, quantum mechanics, statistical mechanics, and thermodynamics, and particularly its relation to entropy. The parametric invariance was used by Ehrenfest as a principle related to the quantization rules of the old quantum mechanics. It was also considered by Rayleigh in the determination of pressure caused by vibration, and the general approach we follow here is based on his. Specific calculation of invariants in classical and quantum mechanics are determined. The Hertz invariant, which is a volume in phase space, is extended to the case of a variable number of particles. We show that the slow parametric change leads to the adiabatic process, allowing the definition of entropy as a parametric invariance.