An adiabatic invariant is a property of a physical system which stays constant when changes are made slowly. In thermodynamics, an adiabatic process is a change that occurs without heat flow and slowly compared to the time to reach equilibrium. In an adiabatic process, the system is in equilibrium at all stages. Under these conditions the entropy is constant. In mechanics, an adiabatic change is a slow deformation of the Hamiltonian where the fractional rate of change of the energy is much slower than the orbital frequency. The area enclosed by the different motions in phase space are then the adiabatic invariants. In quantum mechanics, an adiabatic change is one that occurs at a rate much slower than the difference in frequency between energy eigenstates. In this case, the energy states of the system do not make transitions, so that the quantum number is an adiabatic invariant. The old quantum theory was formulated by equating the quantum number of a system with its classical adiabatic invariant. This determined the form of the Bohr-Sommerfeld quantization rule: the quantum number is the area in phase space of the classical orbit.

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